The power of X

[doronshadmi]

By using a proof by contradiction Cantor proved that Power(X) > X.

Let us carefully examine this proof.

A question:

If –as a part of our proof– we define E and F in such a way that E > F, can we still claim that our proof holds?

Some facts:

1) A set is a collection of distinct objects (each object is included once and only once in a given set).

2) X is a set, and so is Power(X)

3) Let set D be the result of the mapping between Power(X) and X according to this rule:

The members of D are any X member that when mapped with some Power(X) member, this X member is not one of members this Power(X) member, for example:



X Power(X)

a <--> {c,d}
b <--> {a,b}
c <--> {a,b,d,e,f,g,…}
d <--> {d,@,*}
…

In this particular example (out of infinitely many examples) D = {a, c, …}

There are two extreme versions of D which are:

D={} or D={a, b, c, d, …} , where each one of them is the result of infinitely many different mappings between Power(X) and X, but the general thing is this:

D={} only if each X member that is mapped with some arbitrary Power(X) member, is also a member of this arbitrary Power(X) member.

D={a, b, c, d, …} only if each X member that is mapped with some arbitrary Power(X) member, is not a member of this arbitrary Power(X) member.

Between D={} and D={a, b, c, d, …} we can define the infinitely many D results which are not D={} or D={a, b, c, d, …}.

So Cantor's method actually defines Power(X)={{}, … , {a, b, c, d, …}} as a part of a proof that says something about Power(X) (and in this case: Power(X) > X).

Furthermore, the result of how Power(X) is constructed by Cantor, cannot be but
Power(X) > X, because given any D (which is a Power(X) member) there is no X member that is mapped with D.

Some conclusions and questions:

1) We do not need a proof by contradiction in order to conclude that Power(X) > X,
because Cantor actually defines Power(X) (by D's) in such a way that the result cannot be but
Power(X) > X.

2) Is a proof is still a proof if we actually determine the result, as a pert of the proof?

3) Do we need the ZF axiom of the power set (after all Cantor's method actually defines it, without any need of this axiom)?

[drkitten]

By using a proof by contradiction Cantor proved that Power(X) > X.

Let us carefully examine this proof.

A question:

If –as a part of our proof– we define E and F in such a way that E > F, can we still claim that our proof holds?

Since your proof nowhere uses the symbols E or F, their properties are irrelevant, and the proof still holds.

Consider the following:

All men are mortal.
Socrates is a man.
The best flavor of pie is cherry.
Therefore, Socrates is mortal.

The third axiom is irrelevant to the proof, and therefore cannot possibly affect the validity of the underlying argument.


Unfortunately, not only is your question irrelevant to your proof, but your "proof" is irrelevant to Cantor's mathematics. You need at a minimum to define "Power(X)," and if you intend to treat it as a set, you will need to show that it is a set. (That's why you need the Power Set axiom, which defines Power(X) as a set if X is a set.)

[doronshadmi]

Since your proof nowhere uses the symbols E or F, their properties are irrelevant, and the proof still holds.

Consider the following:

All men are mortal.
Socrates is a man.
The best flavor of pie is cherry.
Therefore, Socrates is mortal.

The third axiom is irrelevant to the proof, and therefore cannot possibly affect the validity of the underlying argument.


Unfortunately, not only is your question irrelevant to your proof, but your "proof" is irrelevant to Cantor's mathematics. You need at a minimum to define "Power(X)," and if you intend to treat it as a set, you will need to show that it is a set. (That's why you need the Power Set axiom, which defines Power(X) as a set if X is a set.)

drkitten,

E and F are some placeholders of an idea, and there is no proof here, but only an examination of Cantor's theorem.

If you get that, then please read again my first post.

Thank you.

[nathan]

E and F are some placeholders of an idea

Perhaps if you explained what the idea is that these are placeholders for, that might help?

[drkitten]

Some facts:

1) A set is a collection of distinct objects (each object is included once and only once in a given set).

2) X is a set, and so is Power(X)

3) Let set D be the result of the mapping between Power(X) and X according to this rule:

The members of D are any X member that when mapped with some Power(X) member, this X member is not one of members this Power(X) member, for example:



X Power(X)

a <--> {c,d}
b <--> {a,b}
c <--> {a,b,d,e,f,g,…}
d <--> {d,@,*}
…

In this particular example (out of infinitely many examples) D = {a, c, …}

There are two extreme versions of D which are:

D={} or D={a, b, c, d, …} , where each one of them is the result of infinitely many different mappings between Power(X) and X, but the general thing is this:

D={} only if each X member that is mapped with some arbitrary Power(X) member, is also a member of this arbitrary Power(X) member.

D={a, b, c, d, …} only if each X member that is mapped with some arbitrary Power(X) member, is not a member of this arbitrary Power(X) member.

Between D={} and D={a, b, c, d, …} we can define the infinitely many D results which are not D={} or D={a, b, c, d, …}.

So Cantor's method actually defines Power(X)={{}, … , {a, b, c, d, …}} as a part of a proof that says something about Power(X) (and in this case: Power(X) > X).

Furthermore, the result of how Power(X) is constructed by Cantor, cannot be but
Power(X) > X, because given any D (which is a Power(X) member) there is no X member that is mapped with D.


You know, if someone bet me $1000 that I couldn't write mathematics this badly and this incorrectly, I'm not sure I would be able even deliberately to pack this much misunderstanding and misinformation into a forum post. It's one of those posts where the fundamental errors are almost entirely masked by the superficial and notational errors.

Let's start with the basics.

X is a set, by assumption. The power set of X, Power(X) exists and is also a set -- but this is specifically by the Axiom of Power Sets, so you can't get away with this. D, however, is not a set, but a mapping. And < has no definition until you give it one. The problem isn't that E and F are placeholders, but that they're placeholders for an idea that you fail to express. And, in fact, Cantor's theorem is primarily about the definition of <.

To show you how "real" mathematics works, I will offer a few definitions. A function f from A to B is a subset of AxB such that each element of A occurs at most once in f. A function f is 1:1 if and only if for every A, there is a corresponding B (=f(A)) and if distinct domain elements correspond to distinct range elements (i.e. f(A) = f(A') -> A = A'). A function is onto if for every B, there is an A such that f(A) = B. If I were doing this as a class lecture, I would give lots of examples at this point. Ask if you need them.

Two sets have the same cardinality if there is a function F between the sets that is both 1:1 and onto. If there is no possible 1:1 and onto map between A and B, but there is a 1:1 (but not onto) map from A to B, then we say (with some abuse of notation) that A < B or B>A.

Now, given a set X and the set Power(X), we let D be an arbitrary mapping between them. First, I will show (by constructing a specific D*) that D can be 1:1. Let D* map each element x \in X to the set {x}. {x} is clearly a member of Power(X), and two distinct elements x and y will generate two distincts mapped sets {x} and {y}. Hence X can be mapped 1:1 (but not onto) into the set Power(X)>

Now, I will show that D is never an onto map.

Consider an arbitary element x and its mapping y (i.e. (x,y) \in D). There are two possibilities here -- either x \in y or x \not\in y. Consider the set Z of all x (\in X) for which x \not\in y. (This is clearly a set by the Axiom of separation, and clearly a subset of X,
and hence a member of the power set of X. I claim that for no value of z (\in X) is (z,Z) a member of D. (if there were such an element, then it would introduce a contradiction). Hence D does not include the subset Z in its range and is not onto.

Since D cannot be onto, but can be 1:1, we conclude that X < Power(X).

[doronshadmi]

You know, if someone bet me $1000 that I couldn't write mathematics this badly and this incorrectly, I'm not sure I would be able even deliberately to pack this much misunderstanding and misinformation into a forum post. It's one of those posts where the fundamental errors are almost entirely masked by the superficial and notational errors.

Let's start with the basics.

X is a set, by assumption. The power set of X, Power(X) exists and is also a set -- but this is specifically by the Axiom of Power Sets, so you can't get away with this. D, however, is not a set, but a mapping. And < has no definition until you give it one. The problem isn't that E and F are placeholders, but that they're placeholders for an idea that you fail to express. And, in fact, Cantor's theorem is primarily about the definition of <.

To show you how "real" mathematics works, I will offer a few definitions. A function f from A to B is a subset of AxB such that each element of A occurs at most once in f. A function f is 1:1 if and only if for every A, there is a corresponding B (=f(A)) and if distinct domain elements correspond to distinct range elements (i.e. f(A) = f(A') -> A = A'). A function is onto if for every B, there is an A such that f(A) = B. If I were doing this as a class lecture, I would give lots of examples at this point. Ask if you need them.

Two sets have the same cardinality if there is a function F between the sets that is both 1:1 and onto. If there is no possible 1:1 and onto map between A and B, but there is a 1:1 (but not onto) map from A to B, then we say (with some abuse of notation) that A < B or B>A.

Now, given a set X and the set Power(X), we let D be an arbitrary mapping between them. First, I will show (by constructing a specific D*) that D can be 1:1. Let D* map each element x \in X to the set {x}. {x} is clearly a member of Power(X), and two distinct elements x and y will generate two distincts mapped sets {x} and {y}. Hence X can be mapped 1:1 (but not onto) into the set Power(X)>

Now, I will show that D is never an onto map.

Consider an arbitary element x and its mapping y (i.e. (x,y) \in D). There are two possibilities here -- either x \in y or x \not\in y. Consider the set Z of all x (\in X) for which x \not\in y. (This is clearly a set by the Axiom of separation, and clearly a subset of X,
and hence a member of the power set of X. I claim that for no value of z (\in X) is (z,Z) a member of D. (if there were such an element, then it would introduce a contradiction). Hence D does not include the subset Z in its range and is not onto.

Since D cannot be onto, but can be 1:1, we conclude that X < Power(X).

Thank you for repeating on Cantor's proof by contradiction.

Now hold your horses, and by using what you already know about Cantor's theorem, please read very carefully my first post.

Thank you.

[doronshadmi]

Perhaps if you explained what the idea is that these are placeholders for, that might help?

The general idea that some object is greater than some other object.

[drkitten]

Thank you for repeating on Cantor's proof by contradiction.

Yes. It was my (obviously vain) hope that you would understand it if you saw a slightly different version.


Now hold your horses, and by using what you already know about Cantor's theorem, please read very carefully my first post.

I read it very carefully. It consists, again, of words that you obviously do not understand. In this case, the words include "set," "power," "mapping," and "infinite" --- and a word that you did not use but that is key to the argument, "diagonalization." You also have no idea what > means when applied to sets.

I'm a teacher by profession. I read badly-written student cruft for a living. One thing that I can assure you of --- the reason that no one can extract any meaning from your gibberish is not through lack of reading skills on their part. No amount of reading skill can extract what isn't there. You go beyond badly written student cruft to complete word salad that indicates a complete lack of understanding, not only of the terminology you wish to study, but of the basic ideas and methods of the field.

[Gate2501]

Whoa, Doronshadmi is still making new posts...

If you are so sure that you are right about these things, why not try getting published for a little peer review?

My guess is that somewhere deep down inside you know that you are looney tunes.

[drkitten]

The general idea that some object is greater than some other object.

And in what sense are we using the word "greater"?

[Beerina]

Ok, lemme see if I understand enough to rewrite this.

Given normal definitions of a set and the power set of a set, is the cardinality of a power set always larger than the set itself? Pow(X) > X?


Given an arbitrary mapping between a set and its power set, call it M (assume it is 1:1 and onto, this we seek to contradict) define another set, called D, where this definition holds (note M and D may themselves be represented as sets, specifically of pairs, where the first, x, is an element of X, and the second, y, is an element of Pow(X)):

D is the subset of M {x -> y | x E X and y E Pow(X)} where x is not an element of set y.

His example for this is OK, assuming set X contains each of those symbols.

We know there is an x' in M that maps to D, D being an element of Pow(x).


So consider the element of M that is {x', D}. Is this itself in D?


Well, if x' is an element of one of the members of D, then it cannot be in D.

If x' is not an element of one of the members of D, then it must be in D.

Therefore (?) there is at least one element of Pow(X), namely D, a seemingly reasonably-defined element, that has no corresponding element in X.

Therefore M cannot be constructed. Therefore Pow(X) > X



Don't know if there's a flaw there offhand, but is that about it?

[nathan]

The general idea that some object is greater than some other object.

... and how does claiming that an arbitrary object is greater than some other arbitrary object help prove 'Pow(X) > X'? For suitable definitions of 'greater than' we already know that some objects are greater than some other objects. Here's one such example:
my bike is greater than your bike
see if you can guess a suitable definitions of 'greater than' that makes that a well formed statement :)

[Apathia]

"Cruft" :lol2:
http://cruftbox.com/cruft/docs/cruftdef.html

[doronshadmi]

I'm a teacher by profession.

Maybe, but you are not a real teacher.

Real teachers do not kill the souls of their students by telling them
that they do not understand what they say.

Real teachers are brave enough to be also students as a part of being real teachers, at least in order to understand their students in order to help them to learn by themselves better.

No me and not you and not anyone else on this planet has the authority on wisdom and on different points of view of already known knowledge.

If you have something to say about what I wrote, be brave enough to do it step by step and in details.

This is the only way to show if I am right or wrong, or maybe if there is something new and interesting in what I say, which do not follow the agreed point of view of this subject.

But this:

read it very carefully. It consists, again, of words that you obviously do not understand. In this case, the words include "set," "power," "mapping," and "infinite" --- and a word that you did not use but that is key to the argument, "diagonalization." You also have no idea what > means when applied to sets.

This is the way of a person which is definitely not a real teacher.

[LostAngeles]

That you do not understand what you say is not killing your soul. You're using words whose definitions you have misconstrued and are using them to come to a false conclusion.

As a mathematician in training, I find your notation and word usage to be sloppy and vague. Hence, your OP does not constitute anything resembling a good proof. Please, take Beerina's and drkitten's suggestions and attempt to rewrite it, thank you.

Feel free to use either naive set theory (which you might be ok with) or Zermelo's axiomatic set theory.

[Gate2501]

Real teachers do not kill the souls of their students by telling them
that they do not understand what they say.


Q4kZptthICw

[GreedyAlgorithm]

doronshadmi, you are not a real student. But if you practice being a real student instead of practicing being a bad teacher, you might get better.

[sol invictus]

Real teachers do not kill the souls of their students by telling them
that they do not understand what they say.

Are you drkitten's student? You certainly don't behave that way.

You make nonsensical posts with great confidence on a public educational forum. Part of the purpose of this place is to expose such garbage for what it is.

If you want to learn something, ask - don't declare. Or if you must state nonsense as fact don't whine when it gets labeled as such. If you can't handle it, go somewhere else.

[Apathia]

Doron's a tough nut to crack in part because he's actualy not coming from standard Mathematics. Over the years he's devoloped his own language of "Monadic Mathematics" that borrows terms from a number of different sources, but doesn't use them in precisely the same manner as the place he got them. So it comes to a kind of language confusion.

He's really requiring that we learn his own idiolect (That's a lingusitic term for individual language, not an insult.)
I don't think he realizes how often there's a disconnect between what he thinks he's saying and what a professional mathematician is going to read based on the language.

Mathematics, of course, has numerous dialects and fringe systems. So, we'd have to judge Dorons's system on it own ground.

My curiosty had some time to read one of his .pdfs on his "Monadic Mathematics."
I had an interesting flashback to over 30 years ago when I was in college and tutoring reading and math during the summer.
We used these:
http://www.educationallearninggames.com/cuisenaire-rods.asp
Cuisenaire Rods.
I can well imagine that the manipulation of these is the ultimate source of Doron's mathematical concepts.
He could use these to illustrate for us what he's about.

Now they are excelent tools for teaching basic math in a concrete fashion.
But a Mathematics of Cuisenaire Rods or some pedological equivalent isn't going to reach many of the more abstract concepts contemporary Mathematics is built upon (So Doron's issues with Cantor).

It's clever though, to have formalized a pedological stage in math learning.

BTW, Doron. A teacher who tells you s/he can't understand what you are trying to communicate and demands that you communicate clearly is treating you with compassion and fairness.
Real teachers and real mathematicians are going to be frank with you for your own sake. If you cannot take that and benifit from it, you are your own worst enemy.

Doron, if you haven't seen Cuisenaire Rods in action before, look them up. You'd love them.

[CapelDodger]

Maybe, but you are not a real teacher.

Real teachers do not kill the souls of their students by telling them
that they do not understand what they say.

Did that happen to you? Did a teacher kill your soul? Is that what this is about?

Try to encompass the concept that people can't understand you because you're actually spouting gibberish. Don't ask me explain in detail why it's gibberish, because gibberish is not susceptible to such analysis. It cannot be atomised, it exists only as a whole.

[CapelDodger]

"Cruft" :lol2:
http://cruftbox.com/cruft/docs/cruftdef.html

Cute :). "Cruft" has been added to my lexicon.

[jsfisher]

At the risk of again failing to find meaning among the gibberish, I'll point out this:

If –as a part of our proof– we define E and F in such a way that E > F, can we still claim that our proof holds?


I think doron is trying to say he believes that X < PowerSet(X) is true by definition and therefore no proof was necessary.

[jj]

Maybe, but you are not a real teacher.

Real teachers do not kill the souls of their students by telling them
that they do not understand what they say.

Real teachers are brave enough to be also students as a part of being real teachers, at least in order to understand their students in order to help them to learn by themselves better.

...

This is the way of a person which is definitely not a real teacher.

Hey, fella, that's a really serious professional allegation you're making.

Perhaps you ought to consider a massive, overwhelming apology real soon now?

[Apathia]

I think doron is trying to say he believes that X < PowerSet(X) is true by definition and therefore no proof was necessary.

He's insinuating that there's a "Hidden Asumption" thingy here, he hopes we'll notice for a change, so that he'll finally have an opening we can take into his discussion. Unfortunately it's not one I get.
I think I need the Cuisenaire Rods. :):rolleyes:

[jsfisher]

He's insinuating that there's a "Hidden Asumption" thingy here, he hopes we'll notice for a change, so that he'll finally have an opening we can take into his discussion.


If there is such an assumption, it is well hidden. We shall see. At least doron didn't try to tell us the diagonal proof the reals are non-countable is bogus.

[Little 10 Toes]

Maybe, but you are not a real teacher.

So drkitten is a member of the imaginary teacher set? :confused:

[PixyMisa]

E and F are some placeholders of an idea
The same can be said of all your posts.

[PixyMisa]

I think I need the Cuisenaire Rods. :):rolleyes:
Everyone needs Cuisenaire Rods! When you're tired of learning mathematics, you can build cool towers. :)

At least, that's what we did in the first grade. :D

[doronshadmi]

I think doron is trying to say he believes that X < PowerSet(X) is true by definition and therefore no proof was necessary.

You are in the right direction.

If you know Cantor's proof by contradiction, then please read my first post (which is not a proof of anything but some questions and thoughts about the proof) and air your view about it.

Thank you.

About my question of the necessity of the axiom of the power set, drkitten is right if we are talking about some proof, because our initial terms is that Power(X) is a set, and this initial term is the axiom of the power set.

But Cantor actually gives us (as a part of proof about Power(X)) the exact "recipe" of how to define Power(X) in such a way that cannot be but > X.

This definition of power(X) is actually independent of any proof about power(X), and by using this definition we actually get the set power(X) out of set X, in such a way that cannot be but Power(X) > X, and all this is done (I think) independently of ZF axiom of the power set.

[Complexity]

doronshadmi - I am unwilling to give any of your posts serious consideration.

This is Vanishingly unlikely to change.

I don't think that your ideas have any merit.

Your delusion of a 'hidden assumption' is not even wrong.

I'll check into this thread occasionally to see what others are saying - your nonsense has attracted some good posters in opposition - but your ramblings are of no interest or import.

I wish you'd loose your delusions of grandeur and learn enough mathematics to be able to appreciate the magnficence of what has been accomplished.

[jsfisher]

If you know Cantor's proof by contradiction, then please read my first post (which is not a proof of anything but some questions and thoughts about the proof) and air your view about it.


I am, I have, it isn't, and I will.

Your opening post is rather confused about terminology and the specifics of Cantor's diagonal proof. However, this sentence jumps out at me:

So Cantor's method actually defines Power(X)={{}, … , {a, b, c, d, …}} as a part of a proof that says something about Power(X) (and in this case: Power(X) > X).

(1) It is not a method; it is a proof.
(2) Cantor did not define Power Set as part of his proof.
(3) Nothing in Power Set's definition assures P(S) > S without proof.
(4) The word, "so", implies the statement follows from earlier statements in the post. It does not.

[jsfisher]

But Cantor actually gives us (as a part of proof about Power(X)) the exact "recipe" of how to define Power(X) in such a way that cannot be but > X.

This definition of power(X) is actually independent of any proof about power(X), and by using this definition we actually get the set power(X) out of set X, in such a way that cannot be but Power(X) > X, and all this is done (I think) independently of ZF axiom of the power set.


The Axiom of Power Set defines power set, not Cantor. Here's the axiom:

$$
\forall A \, \exists P \, \forall B \, [B \in P \iff \forall C \, (C \in B \Rightarrow C \in A)]
$$

Quite an elegant definition of P(A), don't you think?

[ArmillarySphere]

Or in other words, P(A) contains all the possible subsets of ? Am I reading that right?
E.g.
P(2) = P({ {{}}, {} }) = { { {{}}, {} }, { {{}} }, { {} }, {} } = { 2, 2\1, 1, 0 }, and so |P(2)| = 4

Nifty!
(edited to fix incorrect numeric equivalences)

[drkitten]

If you have something to say about what I wrote, be brave enough to do it step by step and in details.

I have done, previously in this thread. Just to start with, your failure to define < (or even to recognize that what is needed is a definition of <, not the introduction of two new undefined symbols) renders your work invalid.

if there is something new and interesting in what I say,

There is not. Self-righteous ignorance is neither new nor interesting.

[doronshadmi]

(2) Cantor did not define Power Set as part of his proof.


Yes he did. Power(X) members are actually D sets, and D sets are defined
by Cantor as a part of his proof about Power(X) > X.

[zosima]

1) We do not need a proof by contradiction in order to conclude that Power(X) > X,
because Cantor actually defines Power(X) (by D's) in such a way that the result cannot be but
Power(X) > X.

2) Is a proof is still a proof if we actually determine the result, as a pert of the proof?

3) Do we need the ZF axiom of the power set (after all Cantor's method actually defines it, without any need of this axiom)?

IANAM but...
On #1, I think you're jumping the gun when you say that cantor defines Power(x). The power set is not D but the set of all unique Ds. In other words constructing D is a step in one algorithm for the power set. I think Cantor's definition is very closely related to the recursive algorithm for the power set(you can see it on the wikipedia page for power set). One(non-technical) way of thinking about this is that Cantor makes his proof by looking at the implicit steps in the algorithm for the Power Set and observing that the output must always be larger than the input. There is no problem with this. A careful examination and understanding of the assumptions and details of an operation is exactly what is required to make proofs about an operation.

On #2 he doesn't determine the result, he shows that the result follows from the definition. You wouldn't complain if someone defined a function and made an inductive argument using the function. Then noted that this argument from induction demonstrates a property of the function would you?

On #3 I'm pretty sure you still would need it. One way of looking at the power set axiom is not just that it defines the power set but also that it guarantees the existence and uniqueness of a power set. Making sure this is the case is crucial.

[nathan]

Maybe, but you are not a real teacher.

How is this answering any of drkitten's points? You're just being rude. (I mean, it's not event the true scotsman falacy.)

[jsfisher]

Yes he did. Power(X) members are actually D sets, and D sets are defined
by Cantor as a part of his proof about Power(X) > X.


No, P(X) members are subsets of X. That is a matter of definition and not anything Cantor provided in his proof. The fact that Cantor constructed D in such a way to be a subset of X in no way means D defines P(X).

Here's the essence of Cantor's proof. Given any mapping f(x) from X to P(X), define

$$ D = \{ \, x \in X: x \not \in f(x) \, \} $$

D must be a subset of X, but it can be shown that

$$ \forall x \in X: \, D \neq f(x) $$

and therefore no mapping from X to P(X) can be onto.

No where in there does a definition for P(X) nor its construction appear.

[doronshadmi]

No where in there does a definition for P(X) nor its construction appear.
Wrong, please read the follow:
IANAM but...
On #1, I think you're jumping the gun when you say that cantor defines Power(x). The power set is not D but the set of all unique Ds. In other words constructing D is a step in one algorithm for the power set.

If D was a "single step" (as you claim) where each step is disconnected of any other D "step" (as you claim), then Cantor was not able to use D in order to conclude anything as a part of his proof.

For example, for infinitely many mappings between X and its power set, there is, for example D={}.

Some example (which is no more than a one of infinitely many other examples) of D={} is:

X Power(X)

a <--> {a}
b <--> {b}
c <--> {c}
d <--> {d}
…

and it is clear that D={} cannot be mapped with any of set X members.


Now let us look at the opposite version of D={}, which is D={a,b,c,d,…}

Some example (which is no more than a one of infinitely many other examples) of D={a,b,c,d,…} is:

X Power(X)

a <--> {b}
b <--> {c}
c <--> {d}
d <--> {e}
…

and it is clear that D={a,b,c,d,…} cannot be mapped with any of set X members.


Between D={} and D={a,b,c,d,…} there are the infinitely many D versions that not one of them is D={} or D={a,b,c,d,…}, and these infinitely many D versions are exactly Power(X) members.

In general there is no D version (and any D version is definitely a member of Power(X)) that is mapped with some X member.

In other words, Cantor explicitly defines Power(X) as an inseparable part of a proof that concludes something about Power(X) (and in this case the conclusion is: |Power(X)|>|X|).

I do not think that a proof is a valid proof if we construct our examined objects, and also doing it in such a way that will lead us to some requested result.

[Reality Check]

Wrong, please read the follow:

If D was a "single step" (as you claim) where each step is disconnected of any other D "step" (as you claim), then Cantor was not able to use D in order to conclude anything as a part og his proof.

For example, for infinitely many mappings between X and its power set, there is, for example D={}.

Some example (which is no more than a one of infinitely many other examples) of D={} is:

X Power(X)

a <--> {a}
b <--> {b}
c <--> {c}
d <--> {d}
…

and it is clear that D={} cannot be mapped with any of set X members.


Now let us look at the opposite version of D={}, which is D={a,b,c,d,…}

Some example (which is no more than a one of infinitely many other examples) of D={a,b,c,d,…} is:

X Power(X)

a <--> {b}
b <--> {c}
c <--> {d}
d <--> {e}
…

and it is clear that D={a,b,c,d,…} cannot be mapped with any of set X members.


Between D={} and D={a,b,c,d,…} there are the infinitely many D versions that not one of them is D={} or D={a,b,c,d,…}, and these infinitely many D versions are exactly Power(X) members.

In general there is no D version (and any D version is definitely a member of Power(X)) that is mapped with some X member.

In other words, Cantor explicitly defines Power(X) as an inseparable part of a proof that concludes something about Power(X) (and in this case the conclusion is: |Power(X)|>|X|).

I do not think that a proof is a valid proof if we construct our examined objects, and also doing it in such a way that will lead us to some requested result.

A power set is a set of all subsets of a set.

D is the set of all non-selfish natural numbers (http://en.wikipedia.org/wiki/Cantor's_theorem). It is not a power set.
For example if 1 and 2 were the only non-selfish natural numbers then D = {1,2}. If D was a power set of these then D = { {}, {1}, {2}, {1,2} }.

See the difference?

[doronshadmi]

A power set is a set of all subsets of a set.

D is the set of all non-selfish natural numbers (http://en.wikipedia.org/wiki/Cantor's_theorem). It is not a power set.
For example if 1 and 2 were the only non-selfish natural numbers then D = {1,2}. If D was a power set of these then D = { {}, {1}, {2}, {1,2} }.

See the difference?

No D is a power set, but any D is a member of a power set and no power set exists without its members.

Cantor's proof is based on the existence of members of sets or powersets.

Try to do it by ignoring the members.

$$ D = \{ \, x \in X: x \not \in f(x) \, \} $$

D must be a subset of X and also a member of Power(X), and no D is disconnected from the other D's (or in other words, we define Power(X) as an inseparble part of a proof about Power(X), which is invalid).

[nathan]

I do not think that a proof is a valid proof if we construct our examined objects, and also doing it in such a way that will lead us to some requested result.

Ok, fine. Why do you reject such proofs?

[doronshadmi]

Ok, fine. Why do you reject such proofs?
It is circular.

[nathan]

Cantor's proof is based on the existence of members of sets or powersets.

well duh! it's a proof about sets and power sets. How can one avoid the properties of sets and power sets when discussing the properties of sets and power sets?

[nathan]

It is circular.
In what way is it circular? Is it you don't understand proof by contradiction?

[doronshadmi]

In what way is it circular? Is it you don't understand proof by contradiction?
You have missed it.

We do not need any proof by contradiction here, because in this case Cantor directly provides us a way to define Power(X) in such a way that |Power(X)| > |X|, no matter if X is finite or not.

Furthermore, we do no have to conclude that a non-finite collection of distinct members is complete (we do not have to use the term "for all", but the term "for each" where |Power(X)| > |X| is a permanent state even if no one of the members of X or Power(X) is their final member).

By using this notion we also avoid Cantor's paradox of the power set of all power sets that is greater than itself, and we do not need any proper classes in order to avoid the paradox.

The largest cardinal does not exist because a set is an incomplete object.

At http://forums.randi.org/showthread.php?t=108957 I provided the logical foundations of this incompleteness, which is based on simpler foundations that provide richer and more interesting results, at the moment that they understood.

[nathan]

We do not need any proof by contradiction here, because in this case Cantor directly provides us a way to define Power(X) in such a way that |Power(X)| > |X|, no matter if X is finite or not.
Where did Cantor do this?

Furthermore, we do no have to conclude that a non-finite collection of distinct members is complete (we do not have to use the term "for all", but the term "for each" where |Power(X)| > |X| is a permanent state even if no one of the members of X or Power(X) is their final member).
What do you mean by 'complete', and 'permanent state' here. Also what has the order of set members have to do with anything (IIRC sets are unordered).

By using this notion we also avoid Cantor's paradox of the power set of all power sets that is greater than itself, and we do not need any proper classes in order to avoid the paradox.

Huh? Are you saying that for some sets |Power(X)| ≯ |X| ?

ETA:sigh, once again Doron persists in editing his posts after the fact ...

The largest cardinal does not exist because a set is an incomplete object.
In what way is {1} an incomplete object?

At http://forums.randi.org/showthread.php?t=108957 I provided the logical foundations of this incompleteness.
no, you didn't. You spectacularly failed at convincing anyone and contradicted yourself at several points (where any meaning was apparent).

[doronshadmi]

Where did Cantor do this?
If you have somthing to say about what I wrote, than say it. Please avoid such "questions".


What do you mean by 'complete', and 'permanent state' here. Also what has the order of set members have to do with anything (IIRC sets are unordered).
How is talking here about ordinals? We are talking about Cardinals, and the largest cardinal does not exist ( http://en.wikipedia.org/wiki/Cantor's_paradox ).

About "permanent state", |Power(X)| > |X| no matter how many members there are.


Huh? Are you saying that for some sets |Power(X)| ≯ |X| ?

No at all. |Power(X)| > |X| no matter how many members there are.


In what way is {1} an incomplete object?

It does no have the completeness of Total symmetry at its self state.


no, you didn't. You spectacularly failed at convincing anyone and contradicted yourself at several points (where any meaning was apparent).
Yes I did.

Your abstract ability is too "noisy" and cannot get simplicity at its self state, which is naturally undefined, because any definition is a limitation of the totality (completeness) of simplicity at its self state.

As long as you unaware of this simplicity as the common basis of your thoughts, you will not get it.

[nathan]

If you have somthing to say about what I wrote, than say it. Please avoid such "questions".

ok, so you refuse to say where Cantor 'define[s] Power(X) in such a way that |Power(X)| > |X|, no matter if X is finite or not'.

How is talking here about ordinals? We are talking about Cardinals, and the largest cardinal does not exist ( http://en.wikipedia.org/wiki/Cantor's_paradox ).
Where did ordinals come into this?

About "permanent state", |Power(X)| > |X| no matter how many members there are.

so by 'permanent state' you mean '∀'? Is that it? What about 'complete'?

No at all. |Power(X)| > |X| no matter how many members there are.
ok, so how do we avoid 'Cantor's paradox of the power set of all power sets that is greater than itself'?

It does no have the completeness of Total symmetry at it self state.
oh good grief not that rubbish again. Please define 'completeness of Total symmetry at it self state'.

Yes I did.
yeah right <-- look two assertions combine to make a negation, like wow man!

Your abstract ability is too "noisy" and order to get simplicity at its self state, which is naturally undefined, because any definition is a limitation of the totality (completeness) of simplicity at its self state.

As long as you unaware of this simplicity as the common basis of your thoughts, you will not get it.
gibberish.

[LostAngeles]

well duh! it's a proof about sets and power sets. How can one avoid the properties of sets and power sets when discussing the properties of sets and power sets?

Really.

The first trick of doing a mathematical proof is to either:

A) Go back to axioms you're working from.
B) Go back to the definition.

More often than not, you will go back to the definition. During some studying last night, I found myself repeating this.

"Go back to the definition of divisibility."

"Go back to the definition of divisibility."

"Use the properties."

"Use the properties."

*draws a little square.*

Actually, you will almost always be drawing from the definition or from properties that drop out of the definition with a few short steps, like say that a*0=0 when defining a field (among many, many other structures).

[LostAngeles]

How is talking here about ordinals? We are talking about Cardinals, and the largest cardinal does not exist ( http://en.wikipedia.org/wiki/Cantor's_paradox ).


Not actually a paradox, provided we do not use Naive Set Theory and consider everything to be a set.. RTFWEYLT.

[jsfisher]

No D is a power set, but any D is a member of a power set and no power set exists without its members.

So what?

Cantor's proof starts with an arbitrary set, X. That's a given.

Given X, we know P(X) exists by the Axiom of Power Sets. So we have X and we now have P(X).

We can then image some mapping f(x) from X to P(X) that we will assume to be 1-1 and onto. We are now up to X, P(X), and f(x).

Given f(x), we can define the set, D. That puts us up to X, P(X), f(x), and D. There is no circular reasoning, no loops in definitions.

Cantor's proof is based on the existence of members of sets or powersets.

Well, yeah. Don't sets have members? (A power set is a set, by the way.)

D must be a subset of X and also a member of Power(X), and no D is disconnected from the other D's (or in other words, we define Power(X) as an inseparble part of a proof about Power(X), which is invalid).

No. There is only one set D. It is completely determined by the mapping, f(x). There are no other Ds. Moreover, the fact that D must be a member of P(X) does not define P(X). Quite the opposite. D's membership in P(X) is a direct consequence of the power set's definition.

[jsfisher]

We do not need any proof by contradiction here, because in this case Cantor directly provides us a way to define Power(X) in such a way that |Power(X)| > |X|, no matter if X is finite or not.


Ok, prove it. Cantor's proof is fairly compact. Show us your version of the proof with the shortcuts you suggest exist. No hand waving, no ill-defined rhetoric, just a rigorous proof, please.

[Reality Check]

No D is a power set, but any D is a member of a power set and no power set exists without its members.

Cantor's proof is based on the existence of members of sets or powersets.

Try to do it by ignoring the members.

http://www.randi.org/latexrender/latex.php?$$ D = \{ \, x \in X: x \not \in f(x) \, \} $$
D must be a subset of X and also a member of Power(X), and no D is disconnected from the other D's (or in other words, we define Power(X) as an inseparble part of a proof about Power(X), which is invalid).

Is your D different from the D that is in the Wikipedia article?
where D is defined as "Let D be the set of all non-selfish natural numbers". (http://en.wikipedia.org/wiki/Cantor's_theorem)

If it is can you tell us what proof of Cantor's theorem you are looking at?

[doronshadmi]

Is your D different from the D that is in the Wikipedia article?
where D is defined as "Let D be the set of all non-selfish natural numbers". (http://en.wikipedia.org/wiki/Cantor's_theorem)

If it is can you tell us what proof of Cantor's theorem you are looking at?

My D can be seen in http://forums.randi.org/showpost.php?p=3644296&postcount=39 .

Please read this post this time instead of look at wiki's D.

[jsfisher]

Is your D different from the D that is in the Wikipedia article?
where D is defined as "Let D be the set of all non-selfish natural numbers". (http://en.wikipedia.org/wiki/Cantor's_theorem)

If it is can you tell us what proof of Cantor's theorem you are looking at?

The D, here, is the same as the set B in the wikipedia article under "The Proof" heading. The non-selfish numbers stuff is in the "A detailed explanation" part.

(At least, that's what I meant by it, and doron copy/pasted the Latex from my post.)

[doronshadmi]

Ok, prove it. Cantor's proof is fairly compact. Show us your version of the proof with the shortcuts you suggest exist. No hand waving, no ill-defined rhetoric, just a rigorous proof, please.

http://forums.randi.org/showpost.php?p=3644296&postcount=39

[jsfisher]

http://forums.randi.org/showpost.php?p=3644296&postcount=39

That's not a proof of anything, just a lot of hand waving.

[nathan]

That's not a proof of anything, just a lot of hand waving.
hey, at least you got a sort of answer. All I got was 'don't ask questions' :)

[doronshadmi]

ok, so you refuse to say where Cantor 'define[s] Power(X) in such a way that |Power(X)| > |X|, no matter if X is finite or not'.

That's not a proof of anything, just a lot of hand waving.
It is not a proof but it is a direct way to define |Power(X)| > |X|.

No proof is needed here, and this is exactly my claim.

[jsfisher]

It is not a proof but it is a direct way to define |Power(X)| > |X|.

No proof is needed here, and this is exactly my claim.


(a) You exhibit a collection of sets you call D (with only vague suggestions as to how a given D may have been generated), and you claim each D is a member of P(X).

Ok, so what? 2, 3 and 4 are each members of N, but that doesn't define the set N. Your D sets do not define P(X) either.

(b) You don't get to make up new definitions for already defined things. Power sets are defined (by the Axiom of Power Sets) not by your D sets.

(c) Nothing in your post deduces anything about cardinality.

[nathan]

It is not a proof but it is a direct way to define |Power(X)| > |X|.
How would you adjust your working in order to define |Power(X)| < |X|? After all, if it _is_ a definition, we can define it anyway we want.

[doronshadmi]

(a) You exhibit a collection of sets you call D (with only vague suggestions as to how a given D may have been generated), and you claim each D is a member of P(X).

Ok, so what? 2, 3 and 4 are each members of N, but that doesn't define the set N. Your D sets do not define P(X) either.

(b) You don't get to make up new definitions for already defined things. Power sets are defined (by the Axiom of Power Sets) not by your D sets.

(c) Nothing in your post deduces anything about cardinality.

jsfisher do you remember this?

$$ D = \{ \, x \in X: x \not \in f(x) \, \} $$

D is not a single element but a direct way to constracut Power(X) in such a way that cannot be but |Power(X)| > |X|, no matter how many members are involved (finite or non-finite, it does not matter).

This time please read the all 3 posts one after the other, in order to get the right picture of it:

http://forums.randi.org/showpost.php?p=3638296&postcount=1

http://forums.randi.org/showpost.php?p=3641293&postcount=29

http://forums.randi.org/showpost.php?p=3644296&postcount=39

[Reality Check]

My D can be seen in http://forums.randi.org/showpost.php?p=3644296&postcount=39 .

Please read this post this time instead of look at wiki's D.

It looks like your D is the B in the Wikipedia article:
http://upload.wikimedia.org/math/f/d/c/fdc55bfbf098055688fc13e561e572cf.png

B is not a power set. It is a subset of A such that for all x in A, x ∈ B if and only if x ∉ f(x) where f(x) is a function (mapping) of A to the power set of A.

Your definition of D is "the members of D are any X member that when mapped with some Power(X) member, this X member is not one of members this Power(X) member".
The mapping with some Power(X) member does not mean that the Power(X) member is a member of D - it is a condition on the inclusion of a member of X in D.
Thus D includes members of X but not subsets of X or Power(X) members. Therefore D is not a power set of X.

[jsfisher]

jsfisher do you remember this?

$$ D = \{ \, x \in X: x \not \in f(x) \, \} $$

D is not a single element but a direct way to constracut Power(X) in such a way that cannot be but |Power(X)| > |X|, no matter how many members are involved (finite or non-finite, it does not matter).

Umm, did you fail to notice that P(X) must already be defined in order to define the set D? Remember that f(x) thing? That is an onto mapping we assume exists from X to P(X). You can't turn around, then, and claim to construct P(X) from D, since P(X) had to exist in order to construct D.

In addition, Cantor had only one such set. Exactly one onto mapping f(x) (that is assumed to exist) yields exactly one set D.

But even ignoring all of that, how do you instantly conclude anything about cardinality? You "in such a way that cannot be but" is not a rigorous line of reasoning.

[jsfisher]

This time please read the all 3 posts one after the other, in order to get the right picture of it:


By the way, your continued practice of telling people to reread your earlier posts is rather tiresome. Your prior posts are faulty; the faults have been pointed out; you haven't addressed the faults; there is no point rereading them.

[jsfisher]

How about we work from this as the reference model for Cantor's proof. The details for Steps 1 and 3 are omitted because they are easy enough to provide and don't add much to the "interesting" parts of the proof.

------------------------------------
Let's take as given a standard definition of "A < B" based on one-to-one and onto mappings from A to B.


Assume P(A) is defined by

$$ \forall B \, [B \in P \iff \forall C \, (C \in B \Rightarrow C \in A)]$$


Let's take as given that a one-to-one mapping exists from A to P(A). If no onto mapping from A to P(A) exists, then A < P(A).


Assume an onto mapping exists from A to P(A). Call it f(x).


Define

$$ B = \{ \, x \in A: x \not \in f(x) \, \} $$


Since the members of B are members if A, B must be a member of P(A).


By definition of set B,

$$ \forall x \in A: x \in B \iff x \not \in f(x)$$

Therefore,

$$ \forall x \in A: \, B \neq f(x) $$


Therefore, a member of P(A) is not in the mapping f(x) from A to P(A).


This contradicts the assumption f(x) was an onto mapping. Therefore, A < P(A).

------------------------------------

Now, doron, where in there other than Step #2 is power set defined (or definable)?

Where in there is it a matter of definition that A < P(A)?

[doronshadmi]

Umm, did you fail to notice that P(X) must already be defined in order to define the set D?


Not necessarily.

All we need is X and a map between some X member and some subset of X.

By doing this we get two things in a one ticket:

Thing 1: The collection of D versions is Power(X).

Thing 2: No D version (which is a Power(X) member) can be mapped can be mapped with any one of X members, so |Power(X)| > |X| is inevitable.

[jsfisher]

Not necessarily.

All we need is X and a map between some X member and some subset of X.

That isn't what Cantor did, so what's your point?

[doronshadmi]

Your prior posts are faulty; the faults have been pointed out;

Not at all.

For example, in your last post you write:

"I must take an aside to lay my foundation so we are all on the same page."
( http://forums.randi.org/showpost.php?p=3637338&postcount=777 )

I must say that I completely agree with you, because without a common basis no two thoughts are gathered into a one definition that can be used as a part of our research.

The difference between you and me is this:

You do not provide the common basis that enables two thoughts to be gathered into a one definition, as an explicit essence of your reasoning.

In that case your reasoning is based on the hidden assumption of the common basis.


On the contrary, my reasoning is aware of this common basis as the naturally undefined state of any given definition.

This common basis is naturally undefined because any thought cannot be but a particular aspect of the common basis, where the common basis at its self state is naturally free of any limitation (and a definition is -without a doubt- some categorical limitation).

Since Simplicity itself (the common basis of any given definition) is naturally undefined, we cannot use a definition in order to understand it.

In this case we have no choice but to use an analogy, but we must not mix between simplicity itself (which is naturally undefined) and some analogy of it (which is limited like any other definition).


Here is the analogy, but this time please try no to mix between the naturally undefined (simplicity at its self state) and this particular analogy, which is naturally limited like any other definition:


Simplicity at its self state is like a straight line that has no beginning, no end, and it is an atom (it is not made of any sub elements).

Any definition is like a fold along the straight line.

1. The fold is not a sub-element of the line, but it is a limited thing along the atomic line, where the atomic line is actually the basis of the fold and not vise versa.

2. The fold has no influence on the simplicity of the line as an atomic state, and on its naturally unlimited self state.

At the moment that 1 and 2 are directly understood (without any need of some fold (analogy, definition, thought, etc. ) as the simplest and naturally undefined basis of your consciousness, then and only then you are able to understand my framework.

Another analogy (which is, again, a naturally limited thing):

I try to show you a way to get a direct experience of a sea without waves, and you try to get it by making waves. By making waves, you will never get it because any wave is exactly not a sea without waves.

At the moments that you get the sea without waves, you are enable to get any wave right from the common state of any other wave, which is the sea without waves (the naturally non-limited atomic (and naturally undefined) state of any limited (and naturally defined) thing).

[nathan]

In that case your reasoning is based on the hidden assumption of the common basis.

I think the underlying assumption everyone but you is making, is the desire to communicate effectively. You do not seem to want to communicate, yet you post here. To what purpose?

[doronshadmi]

That isn't what Cantor did, so what's your point?
My point is this:

By providing D, Cantor does not need any proof that |Power(X)| > |X| and he also does not need to use the axiom of the power set, because he directly constructs Power(X) from X by providing D.

Simple and beautiful, isn't it?

[ArmillarySphere]

Not necessarily.

All we need is X and a map between some X member and some subset of X.
So... we have X and some f(x) that produces subsets of X

It is trivial to show that at least one of these exists, e.g. f(x) = {x}, or even f(x) = {}. (i.e, constant). The problem is that without any additional conditions on the mapping, you're going to get a far larger variation of f(x) mappings than you have of actual D versions. In other words, the second part:

By doing this we get two things in a one ticket:

Thing 1: The collection of D versions is Power(X).
This is tautological - D versions are subsets of X, and Power(X) is the collection of all possible subsets of X. No need for the mappings to know this.

Thing 2: No D version (which is a Power(X) member) can be mapped can be mapped with any one of X members, so |Power(X)| > |X| is inevitable.
In other words, if we take F = the set of all possible f(x), then |F| > |X|. Quite, but it's also true that |F| > |Power(X)|, and this says nothing about the cardinalities of Power(X) and X. Just by shuffling the mapping a bit, I can easily map any arbitrary D to any member of X - and I've only treated *one* possible D here.

[nathan]

Since Simplicity itself (the common basis of any give definition) is naturally undefined, we cannot use a definition in order to understand it.
If you cannot provide a definition for a term you use, the term is meaningless. Why have you chosen to use the word 'Simplicity' for this undefinable thing? Why not use the word 'Potato'?

In this case we have no choice but to use an analogy, but we must not mix between simplicity itself (which is naturally undefined) and some analogy of it (which is limited like any other definition).
Incorrect. If 'simplicity' (is that different from 'Simplicity'? I can't tell, because you fail to define the two terms), is undefined, how can one know any analogy is accurate?

Simplicity at its self state is like a straight line that has no beginning, no end, and it is an atom (it is not made of any sub elements).

Why isn't Simplicity like a point? that has no beginning, no end and is not made of sub elements? What space is this straight line embedded in? Does it matter?

Any definition is like a fold along the straight line.
In what way can a straight line have a fold in it?

1. The fold is not a sub-element of the line, but it is a limited thing along the atomic line, where to atomic line is actually the basis of the fold and not vise versa.
Gibberish
2. The fold has no influence on the simplicity of the line as an atomic state, and on its naturally unlimited self state.
Gibberish
At the moment that 1 and 2 are directly understood (without any need of some fold (analogy, definition, thought, etc. ) as the simplest and naturally undefined basis of your consciousness, then and only then you are able to understand my framework.
Given that #1 and #2 are gibberish, I conclude your framework is gibberish.

I try to give you way to get a direct experience of a see[sea?] without waves, and you try to get it by making waves. By making waves, you will never get it because any wave is exactly not a sea without waves.
A calm sea is not exactly the same as a perturbed sea. ok.

At the moments that you get the sea without waves, you are enable to any wave from the common state of any other wave, which is the sea without waves (the naturally non-limited atomic (and naturally undefined) state of any limited (and naturally defined) thing).
This is not a sentence. Ergo, gibberish.

[doronshadmi]

I think the underlying assumption everyone but you is making, is the desire to communicate effectively. You do not seem to want to communicate, yet you post here. To what purpose?
Dear Nathan,

If this time you will read all of http://forums.randi.org/showpost.php?p=3647417&postcount=70, you will find that you and me are talking on the same thing, communication.

I simply take this concept from its simplest state.

[nathan]

I simply take this concept from its simplest state.
No, you don't. You use sequences of words that have no meaning. That is not communication.

and to reiterate jsfisher's point. I'm getting tired of you continually telling people to reread your posts. Explain what you mean, or go away [to co-opt a well used phrase at the JREF]

[doronshadmi]

In other words, if we take F = the set of all possible f(x), then |F| > |X|. Quite, but it's also true that |F| > |Power(X)|, and this says nothing about the cardinalities of Power(X) and X. Just by shuffling the mapping a bit, I can easily map any arbitrary D to any member of X - and I've only treated *one* possible D here.

In that particular case F = Power(Power(X)) and |F| > |Power(X)|.

But I am talking about the genaral case of |Power(X)| > |X| where X is a placeholder of any set (and a powerset is a set (a collection of distecnt objects, where order is not important)).

[doronshadmi]

No, you don't. You use sequences of words that have no meaning. That is not communication.

Communication right from silence itself is the best communication, because any sound (definition) is totally clear when observed from silence itself.

No sound (definition, which is limited by nature) is a natural basis for real communication.

It can be understood only if you directly get silence itself as the simplest state of any thought.

Your struggle to find a common basis at the level of sound (definition) actually makes you tired.

[jsfisher]

My point is this:

By providing D, Cantor does not need any proof that |Power(X)| > |X| and he also does not need to use the axiom of the power set, because he directly constructs Power(X) from X by providing D.

Cantor could have done many, many things, but he didn't construct the power set of X by providing D. He didn't provide the collection of sets D you claim. He needed exactly one set of that sort, so he exhibited only one

Go back to the version of Cantor's proof I posted. Please point out in it where you think the power set is "constructed" from the set D (named B in my posting).

Simple and beautiful, isn't it?

Cantor's proof? Yes, it is.

Your approach, on the other hand, is long and contorted. Moreover, you are supposed to be proving that A < P(A), so you must use the Axiom of the power set to give meaning to P(A).

The long and contorted part is where you'd need to prove that P(A) can be derived from all your D sets. You haven't done that.

[ArmillarySphere]

In that particular case F = Power(Power(X)) and |F| > |Power(X)|.
Flatly incorrect. The functions f(x) are not subsets of Power(X) - even you should see that.

My point was that just because any "D version (which is a Power(X) member) [...] can be mapped with any one of X members", i.e. there are more possible f(x) definitions than members of X, this implies nothing about the cardinalities of Power(X) and X.


By the way, I had to paraphrase to make sense of the original post - your grammar is so bad that the original statement makes no sense at all.

This is what you wrote:
Thing 2: No D version (which is a Power(X) member) can be mapped can be mapped with any one of X members
If you meant that no D version can be mapped with any one X member, then D violates its own definition.
If you meant that any D version can be mapped with any one X member, well, that follows from the definition of f(x). And as noted above, implies nothing for the collection of possible D versions.

[jsfisher]

Not at all.

For example, in your last post you write:...


I see the dearth of your Mathematical aptitude is exceeded by the enormity of your sense of self-importance.

[doronshadmi]

I see the dearth of your Mathematical aptitude is exceeded by the enormity of your sense of self-importance.
So you did not read http://forums.randi.org/showpost.php?p=3647417&postcount=70 .

If you read it, you will find that silence at its self state is the most humble state of mind, because any thought (which is limited by nature) gets its most fulfilled state if based on the simplest state of mind.

[ArmillarySphere]

By the way, the way you've defined Power(X) does not yield the same result as is outlined in the original axiom, so I don't see how any proof you base on that is valid.

Try the empty set: X = {}
Now, there are *no* mappings that will yield a set D = {} from any member of X (since there are none), thus according to that definition, Power({}) = {}

Not only does that violate what we were trying to prove, it's also wrong. Power({}) is actually {{}}

[doronshadmi]

My point was that just because any "D version (which is a Power(X) member) [...] can be mapped with any one of X

Not at all. No D version (which is a subset of X and a member of Power(X)) is mapped with any one of X members, and this is exactly the reason why |Power(X)| > |X|.

[doronshadmi]

By the way, the way you've defined Power(X) does not yield the same result as is outlined in the original axiom, so I don't see how any proof you base on that is valid.

Try the empty set: X = {}
Now, there are *no* mappings that will yield a set D = {} from any member of X (since there are none), thus according to that definition, Power({}) = {}

Not only does that violate what we were trying to prove, it's also wrong. Power({}) is actually {{}}

X = {} and Power(X)={{}}.

Please think general. D is not any particular result, but it is always a mamber of Power(X) that is not mapped with any member of X.

That's all. If you get it, you get what I claim.

[nathan]

Communication right from silence itself is the best communication, because any sound (definition) is totally clear when observed from silence itself.

No sound (definition, which is limited by nature) is a natural basis for real communication.

It can be understood only if you directly get silence itself as the simplest state of any thought.

Your struggle to find a common basis at the level of sound (definition) actually makes you tired.

Are you saying Mark Twain was right?
It is better to keep your mouth shut and appear stupid than to open it and remove all doubt.
If not, what are you saying? That communication should be silent? Text-based communication is inherently silent -- unless you count the non-information-carrying noise of keyboard clatter.

Oh, and what has this to do with your contention that Cantor's proof of |X| < |Power(X)| is over-complicated?

[Don At Work]

So you did not read http://forums.randi.org/showpost.php?p=3647417&postcount=70 .

If you read it, you will find that silence at its self state is the most humble state of mind, because any thought (which is limited by nature) gets its most fulfilled state if based on the simplest state of mind.

Word salad.
Not even wrong.
:deadhorse

This is not mathematics.
You have been refuted on nearly EVERY SINGLE POST. Not contradicted, REFUTED. You either ignore the evidence, or simply say to re-read an earlier post. You simply (should I even use that word?) don't get it.

[jsfisher]

So you did not read http://forums.randi.org/showpost.php?p=3647417&postcount=70 .

If you read it, you will find that silence at its self state is the most humble state of mind, because any thought (which is limited by nature) gets its most fulfilled state if based on the simplest state of mind.


Stay on topic, please. The above is material for a different thread.

[ArmillarySphere]

X = {} and Power(X)={{}}.

Please think general. D is not any particular result, but it is always a mamber of Power(X) that is not mapped with any member of X.

That's all. If you get it, you get what I claim.

Quite, so D is indeed *not* just any subset of X, but defined using a mapping which requires Power(X) in order to be defined at all. Which you claimed wasn't necessary before.

Still following Cantor, still using proof by contradiction, not defining Power(X) along the way as you claim. The only difference is that Cantor's version was a lot easier to follow.

[doronshadmi]

Word salad.
Not even wrong.
:deadhorse

This is not mathematics.
You have been refuted on nearly EVERY SINGLE POST. Not contradicted, REFUTED. You either ignore the evidence, or simply say to re-read an earlier post. You simply (should I even use that word?) don't get it.
Yes I was refuted by people like you that are unaware of the common basis of their thoughts, which is itself not a thought, but the simplest state of consciousness, which is the natural basis of any thought (and since a definition is some thought, it is also the natural basis of any definition).

Also a Computer stuff (or any other abstract or non-abstract mechanical method) is nothing but some agent of your consciousness.

In other words, real mathematician is first of all a person that is aware of the simplest state of consciousness as an inseparable part of his mathematical work.

[doronshadmi]

Quite, so D is indeed *not* just any subset of X, but defined using a mapping which requires Power(X) in order to be defined at all. Which you claimed wasn't necessary before.

Still following Cantor, still using proof by contradiction, not defining Power(X) along the way as you claim. The only difference is that Cantor's version was a lot easier to follow.
Do you really need the axiom of the power set in order to gather infinitely many Ds (X subsets, that no one of them is mapped with any X member, because of D's definition) into a one set?


By D's definition we have D={}, D={a, b, c, d, ...} and any version between them (and no D version is mapped with any of the X members).

So, as I said, all is needed is D's definition, and we get both Power(X) and |Power(X)| > |X|, without any proof, but simply by direct construction.

[ArmillarySphere]

Firstly, assuming such sets D exist at all, which isn't clear, this would be a very roundabout way of defining the power set of X as "all possible subsets of X", i.e. the standard definition.

Secondly, how do you get to "that no one of them is mapped with any X member"? Unless you have a mapping from X to Power(X) in the first place, there's no way to get at the "remainder" set D at all.

[doronshadmi]

Firstly, assuming such sets D exist at all, which isn't clear, this would be a very roundabout way of defining the power set of X as "all possible subsets of X", i.e. the standard definition.

Secondly, how do you get to "that no one of them is mapped with any X member"? Unless you have a mapping from X to Power(X) in the first place, there's no way to get at the "remainder" set D at all.

Please, please read all of http://forums.randi.org/showpost.php?p=3645862&postcount=63 (including the links), it is all there.

[ArmillarySphere]

Where does your f(x) come from?

[drkitten]

Do you really need the axiom of the power set in order to gather infinitely many Ds (X subsets, that no one of them is mapped with any X member, because of D's definition) into a one set?

Finally, a genuine question.

Yes, you do. The power set axiom is independent of the other axioms of ZFC.

Here's an article (http://www.jstor.org/sici?sici=0002-9890(196908%2F09)76%3A7%3C787%3AOTIOST%3E2.0.CO%3B 2-7&cookieSet=1) discussing why in detail.



By D's definition we have D={}, D={a, b, c, d, ...} and any version between them (and no D version is mapped with any of the X members).

Nope. You can't prove that without using the Power Set Axiom.

[doronshadmi]

Finally, a genuine question.

Yes, you do. The power set axiom is independent of the other axioms of ZFC.

Here's an article (http://www.jstor.org/sici?sici=0002-9890(196908%2F09)76%3A7%3C787%3AOTIOST%3E2.0.CO%3B 2-7&cookieSet=1) discussing why in detail.




Nope. You can't prove that without using the Power Set Axiom.
Please provide the right link to this articale, because I do not see it.

Thank you.

[doronshadmi]

Where does your f(x) come from?


Let f be any function from X into the subsets of X according to this rule:


$$ D = \{ \, x \in X: x \not \in f(x) \, \} $$


In other owrds, D's versions are exactly Power(X) and no X member is mapped with any D version.

To claim that we have no guarantee that any version of D really exists between D={} and D={a, b, c, d, …} is similar to the claim that there are R members that are greater than 0 and smaller than 1, that there is no guarantee that they are between 0 and 1.

[drkitten]

Please provide the right link to this articale, because I do not see it.

Thank you.

The link is correct; you may simply not have access to the JSTOR archives.

The paper article can be found at:

On the Independence of Set Theoretical Axioms
Alexander Abian
The American Mathematical Monthly, Vol. 76, No. 7 (Aug. - Sep., 1969), pp. 787-790

... which should be in any decent university library.

Quoting from the relevant paper (apologies for typos):

To prove that the axiom of Power-set is independent of the remaining three axioms, it is enough to consider a model whose domain consists of a single set s, where

(3) s \not\in s

i.e s is an empty set. Clearly, in this model the axiom of Extensionality is valid. However, since in this model there is no set whose element is s, the axiom of Power-set is not valid. On the other hand, since in this model Us = s, and since s is its own selection set, the axiom of sum-set as well as the axiom of Choice is [sic] valid.

[jsfisher]

Let f be any function from X into the subsets of X according to this rule:


$$ D = \{ \, x \in X: x \not \in f(x) \, \} $$


In other owrds, D's versions are exactly Power(X) and no X member is mapped with any D version.

Enough with your sloppy Mathematical language and hand waving. You said "let f be any function...". That means only one (although it is arbitrary as to which one). Did you really mean to speak of a family of functions, fi? That is,

$$ D _{i} = \{ \, x \in X: x \not \in f _{i} (x) \, \} $$

Now, just how to construct the power set of X from the Di sets? (i.e. "D's versions")?

How are you going to show that your construction is in fact the power set of X without reference to the Axiom of Power Sets?

[drkitten]

How are you going to show that your construction is in fact the power set of X without reference to the Axiom of Power Sets?

Oooooh, I know this one.

She's going to tell us to re-read irrelevant posts and then snidely tell us that it's an immediate consequence of some ill-defined term that we fail to understand because we are unaware of the fundamental unity of the positronium windings of the verteron coils or some such Star Trek technobabble.

And then she's going to insult people who actually read her pseudomathematics closely enough to correct some of the more gross mistakes and misstatements.

[ArmillarySphere]

Doesn't the axiom merely state that "yes, every set has a power set"?

I think of the definition and axiom separately - one thing just explains what we're talking about and the other asserts some property of it. You still need the definition, however...

[drkitten]

Doesn't the axiom merely state that "yes, every set has a power set"?

And that the power set exists and is itself a set.

See my previous posting, which cites a simple model where the "power set" does not exist (more accurately, the power set would be a proper class, not a set).

[jsfisher]

And that the power set exists and is itself a set.

See my previous posting, which cites a simple model where the "power set" does not exist (more accurately, the power set would be a proper class, not a set).

It also defines exactly what a power set is, i.e. set of all subsets of a set, doesn't it?

[doronshadmi]

Enough with your sloppy Mathematical language and hand waving. You said "let f be any function...". That means only one (although it is arbitrary as to which one). Did you really mean to speak of a family of functions, fi? That is,

$$ D = \{ \, x \in X: x \not \in f (x) \, \} $$

Now, just how to construct the power set of X from the Di sets? (i.e. "D's versions")?

How are you going to show that your construction is in fact the power set of X without reference to the Axiom of Power Sets?

Function here means mapping. So any function means any mapping from X into the subsets of X .

By using $$ D = \{ \, x \in X: x \not \in f (x) \, \} $$ you constract the power set of X and also show that |Power(X)| > |X|.

[ArmillarySphere]

And that the power set exists and is itself a set.

See my previous posting, which cites a simple model where the "power set" does not exist (more accurately, the power set would be a proper class, not a set).
Quite, I missed that one. Fair enough - I'm not abstract enough in my thinking it seems, since I usually limit my examples to useful domains :p

[ArmillarySphere]

Function here means mapping. So any function means any mapping from X into the subsets of X .
As I thought - you are using P(X), despite claiming the opposite. "The subsets of X" has no meaning unless that's what you are in fact speaking of - without the set defined as the "other end" of the mapping, there's nothing preventing all members of X to map to the same value. And as the link drkitten demonstrated, P(X) isn't even a set in the first place without the Axiom of Power Set.

Circularity, gone. What's next? Oh, right - the D

doron, you're overachieving here. It's not necessary to iterate over all possible D's here; you'd just be repeating the definition of P(X) again. All that's necessary is to show that there's a single one. But of course, that requires you to use the defined meanings of > and onto mappings... and rigorous definitions aren't really your thing, right?

[jsfisher]

By using...you constract the power set of X and also show that |Power(X)| > |X|.


Prove it.

[doronshadmi]

As I thought - you are using P(X), despite claiming the opposite. "The subsets of X" has no meaning unless that's what you are in fact speaking of - without the set defined as the "other end" of the mapping, there's nothing preventing all members of X to map to the same value. And as the link drkitten demonstrated, P(X) isn't even a set in the first place without the Axiom of Power Set.

Circularity, gone. What's next? Oh, right - the D

doron, you're overachieving here. It's not necessary to iterate over all possible D's here; you'd just be repeating the definition of P(X) again. All that's necessary is to show that there's a single one. But of course, that requires you to use the defined meanings of > and onto mappings... and rigorous definitions aren't really your thing, right?
Since we are talking here about sets and a set is a collection of distinct members (where order is not important) then each D version is unique even if there are infinitely many mappings between X members and X subsets that their results are the same D version. In other words, we care only about the unique cases, and the collection of the unique D versions is exactly the power set of X.

There is nothing to prove here because it is a direct construction of the power set of X.

Furthermore, this construction is done in such a way that the result cannot be but
|Power(X)| > |X| and we do not have to use our brain too much in order to immediately clearly get it.

Since no set is a complete object, when compared to Total Symmetry, then the whole ugly notion of proper classes is avoided.

[jsfisher]

Since we are talking here about sets and a set is a collection of distinct members (where order is not important) then each D version is unique even if there are infinitely many mappings between X members and X subsets that their results are the same D version. In other words, we care only about the unique cases, and the collection of the unique D versions is exactly the power set of X.

Prove it.

There is nothing to prove here because it is a direct construction of the power set of X.

No, the power set is defined by the Axiom of Power Set. You'd need to prove you've managed to enumerate the necessary ones in order to claim "the collection...is exactly the power set." by the way, you left out a very important word, union, in there.

Curiously, you had been claiming you didn't need the definition of power set; you'd define it by its construction. My, how times change.

Furthermore, this construction is done in such a way that the result cannot be but |Power(X)| > |X| and we do not have to use our brain too much in order to immediately clearly get it.

Mathematical proofs clear do not depend on your use of your brain. Be that as it may, your repeated "the result cannot be but..." falls short of any sort of proof.

Since no set is a complete object, when compared to Total Symmetry, then the whole ugly notion of proper classes is avoided.

Off-topic gibberish.

[jsfisher]

Enough with your sloppy Mathematical language and hand waving. You said "let f be any function...". That means only one (although it is arbitrary as to which one). Did you really mean to speak of a family of functions, fi? That is,

$$ D = \{ \, x \in X: x \not \in f (x) \, \} $$

Now, just how to construct the power set of X from the Di sets? (i.e. "D's versions")?
How are you going to show that your construction is in fact the power set of X without reference to the Axiom of Power Sets?

Function here means mapping. So any function means any mapping from X into the subsets of X .

By using $$ D = \{ \, x \in X: x \not \in f (x) \, \} $$ you constract the power set of X and also show that |Power(X)| > |X|.


Doron, you edited my post without out any indication of what or why.

Basically, you have attributed text from a post to me, yet, that isn't my text, because you changed it. You are being very dishonest.

Why, doron? Why are you lying?

[LostAngeles]

What makes this great is all the whining I've ever done about graders requiring me to be very exact in my proofs and what degree of exactness I need and when can I just say, "It's trivial."

I <3 the comedy gold going on here.

[drkitten]

Since we are talking here about sets and a set is a collection of distinct members (where order is not important)

Well, here's another problem. You're using naive set theory without realizing it. A set is indeed a collection of distinct members -- but not every collection of distinct members is a set. For example, the collection of all (distinct) sets is not a set, but a proper class.

You're fortunate in a sense; most people who try to apply the theorems of axiomatic set theory to informal naive concepts of sets get paradoxes. You just get gibberish.

[drkitten]

It also defines exactly what a power set is, i.e. set of all subsets of a set, doesn't it?

Technically, you have just offered a definition of "power set" yourself without using the power set axiom.

However, the definition by itself doesn't give you enough traction. In order to manipulate the power set of an arbitrary set X, you also need to show a few things.

First, that the power set exists.
Second, that the power set is itself a set.
Third, that the power set is unique.

The first two are given by the power set axiom. The third is given by the power set axiom plus the axiom of extensionality.

(Think of it this way : I could define Oberon as the king of the fairies, but that doesn't mean that I can do anything useful unless I can establish that fairies exist and that they have a king.)

[doronshadmi]

For example, the collection of all (distinct) sets is not a set, but a proper class.
"all" means "complete" or "total".

Since a collection is not a total thing (where total is naturally undefined) "the collection of all blablabla" is nothing but a self contradiction.

Your limited notion is based on limited notions. It is circular because you use thoughts in order to define thoughts (any definition is first of all a thought).

As long as your reasoning is closed under thoughts (where one thought holds the tail of another thought) you will not able to directly get the naturally undefined (Simplicity itself, which is the common ground of any thought, but it is not itself a thought).

The thing that is not a set, is also not a collection and it is the non-local atom which is the complete naturally undefined and unlimited common ground of any collection (finite or not, of distinct members or not).

Your circular closed under thoughts reasoning is only an illusion, and therefore it is gibberish from top to bottom.

The sad fact is the your poor students have to eat your gibberish in order to get their diploma, which is the "closed under thoughts" diploma.

[doronshadmi]

Doron, you edited my post without out any indication of what or why.

Ho sorry, Your are right.

I edited it in order you use it as a part of my post and by mistake I also put it as a part of your quote.

Here is the right one:

Enough with your sloppy Mathematical language and hand waving. You said "let f be any function...". That means only one (although it is arbitrary as to which one). Did you really mean to speak of a family of functions, fi? That is,

$$ D _{i} = \{ \, x \in X: x \not \in f _{i} (x) \, \} $$

Now, just how to construct the power set of X from the Di sets? (i.e. "D's versions")?

How are you going to show that your construction is in fact the power set of X without reference to the Axiom of Power Sets?

and my answer was that "any function" means "any mapping".

[doronshadmi]

Prove it.

This time please read http://forums.randi.org/showpost.php?p=3648068&postcount=97 until the end of it.

Thank you.

[doronshadmi]

Here is the wiki version of the axiom of the power set ( http://en.wikipedia.org/wiki/Axiom_of_power_set ):

"Given any set A, there is a set P(A) such that, given any set B, B is a member of P(A) if and only if B is a subset of A."


First, that the power set exists.
Second, that the power set is itself a set.
Third, that the power set is unique.

In other words, if we have a way to construct {}, {a,b,c,d,...} and anything between them, then we get P(A), and this is exactly what Cantor did by using: $$ D = \{ \, x \in X: x \not \in f (x) \, \} $$

"(Subset is not used in the formal definition above because the axiom of power set is an axiom that may need to be stated without reference to the concept of subset.)"

The notion of proper classes do not hold because no collection is a total (complete) thing. "The collection of all ..." DO NOT HOLD WATER, at the moment that your notion is not a "closed under thoughts" framework.

[ArmillarySphere]

"all" means "complete" or "total".

Since a collection is not a total thing (where total is naturally undefined) "the collection of all blablabla" is nothing but a self contradiction.
No, all means all. Don't introduce your vaguely defined ideolect into this.

According to your logic, the set of natural numbers does not exist, since it's the set of all natural numbers...

Your limited notion is based on limited notions. It is circular because you use thoughts in order to define thoughts (any definition is first of all a thought).

As long as your reasoning is closed under thoughts (where one thought holds the tail of another thought) you will not able to directly get the naturally undefined (Simplicity itself, which is the common ground of any thought, but it is not itself a thought).
Incomprehensible non sequitur

The thing that is not a set, is also not a collection and it is the non-local atom which is the complete naturally undefined and unlimited common ground of any collection (finite or not, of distinct members or not).
Flatly wrong. There are a lot of proper classes - in brief, every collection that isn't a set has to be one. You yourself insisted on the collection of all distinct sets being one (which is fine, by the way - it's merely defining the domain).

Your circular closed under thoughts reasoning is only an illusion, and therefore it is gibberish from top to bottom.

The sad fact is the your poor students have to eat your gibberish in order to get their diploma, which is the "closed under thoughts" diploma.
Yes, how horrid that students have to learn to explain their reasoning...

[doronshadmi]

No, all means all. Don't introduce your vaguely defined ideolect into this.

According to your logic, the set of natural numbers does not exist, since it's the set of all natural numbers......

Don't use your "closed under thoughts" circular reasoning, in order to understand my framework.

According to my Logic there exists the set of Natural numbers, but like any collection, it is incomplete (something that cannot be understood by a "closed under thoughts" circular reasoning).

Incomprehensible non sequitur
You are right, because your abstract ability is "closed under thoughts" .

Yes, how horrid that students have to learn to explain their reasoning...
Very simple. Any student is first of all directly aware of the simplest state of consciousness, which is itself not a thought but the unbounded atom naturally undefined common ground of any thought (and again, any definition is first of all a thought).

By using the non-researchable as the common ground of the researchable we get a circular-free framework.

This simple beauty cannot be understood by any person that uses a "closed under thoughts" circular reasoning.

By using a circular-free framework, every student will be able the share his unique point of view in such a way that will be understood by the rest of the students, because they share the same common ground, which is the natural basis of any thought (any definition).

I know that you do not understand a single word of what I say because you try to understand it by a "closed under thoughts" method.

[drkitten]

This time please read http://forums.randi.org/showpost.php?p=3648068&postcount=97 until the end of it.

Thank you.

Oooh, can I call it or what?

I wrote:


She's going to tell us to re-read irrelevant posts

Which she did:


This time please read http://forums.randi.org/showpost.php...8&postcount=97 until the end of it.


I then wrote that :

and then snidely tell us that it's an immediate consequence of some ill-defined term


and she then wrote:

"all" means "complete" or "total".

Since a collection is not a total thing (where total is naturally undefined) "the collection of all blablabla" is nothing but a self contradiction.

I then predicted that she would say ...

that we fail to understand because we are unaware of the fundamental unity of the positronium windings of the verteron coils or some such Star Trek technobabble.

just as she did here:


By using the non-researchable as the common ground of the researchable we get a circular-free framework.

This simple beauty cannot be understood by any person that uses a "closed under thoughts" circular reasoning.

By using a circular-free framework, every student will be able the share his unique point of view in such a way that will be understood by the rest of the students, because they share the same common ground, which is the natural basis of any thought (any definition).

I know that you do not understand a single word of what I say because you try to understand it by a "closed under thoughts" method.

and finally, I predicted that

And then she's going to insult people who actually read her pseudomathematics closely enough to correct some of the more gross mistakes and misstatements.

Which she fulfilled by writing


Your limited notion is based on limited notions. It is circular because you use thoughts in order to define thoughts (any definition is first of all a thought).

As long as your reasoning is closed under thoughts (where one thought holds the tail of another thought) you will not able to directly get the naturally undefined (Simplicity itself, which is the common ground of any thought, but it is not itself a thought).

[snip]

Your circular closed under thoughts reasoning is only an illusion, and therefore it is gibberish from top to bottom.

The sad fact is the your poor students have to eat your gibberish in order to get their diploma, which is the "closed under thoughts" diploma.

Can I call it, or what? I'd like my million now, Mr. Randi....

[Third Eye Open]

:clap: Bravo Drkitten, you must have some very powerful mind powers!

I know pretty much zilch about mathematics such as this, but I find this quote hillarious!

Your limited notion is based on limited notions. It is circular because you use thoughts in order to define thoughts (any definition is first of all a thought).

It all makes sense now! Doron does his/her math without the use of thoughts!!

[drkitten]

It all makes sense now! Doron does his/her math without the use of thoughts!!

"Simplicity itself!"

I'll leave you to guess whether the OED defintion A.10 or B.6 is the more appropriate "simple" here.

[drkitten]

I know pretty much zilch about mathematics such as this,

It's not that bad. Basically, this is the difference between "naive" set theory, which is understandable but broken, and "axiomatic" or "formal" set theory, which works but is incomprehensible.

In naive set theory, you can create a "set" of objects that satisfy any given property, such as the "set of all sets" or the "set of all things that aren't Britney Spears" or the "set of all sets that contain themselves" or the "set of all sets that do not contain themselves."

The problem is, if you try to reason about the last one (specifically, is it a member of itself or not) you get into a contradiction.

Axiomatic set theory fixes this by saying you have to build sets from the ground up, not from the top down. So if you have two sets A and B, you can make A U B or {A,B}. The Power Set axiom just is one way to make big sets out of small sets.

[doronshadmi]

Can I call it, or what? I'd like my million now, Mr. Randi....

Why, because you are "closed under thoughts" circular reasoning?

You even do not aware of how two thoughts of you a gathered into a one idea, so go claim on another tree, where a definition (which is nothing but a thought) is the basis of your "closed under thoughts" circular reasoning.

[jsfisher]

Technically, you have just offered a definition of "power set" yourself without using the power set axiom.

Yes, I knew I'd done that....it was just wanting to be sure my understanding of the Axiom of Power Set was correct...that it defined what was meant by power set (in addition to its other properties).

[jsfisher]

This time please read http://forums.randi.org/showpost.php?p=3648068&postcount=97 until the end of it.

Thank you.

No, really, prove it.

(By the way, since we are at the level of set theory, you cannot use properties of reals in any proof since they are at a much higher level in the mathematical hierarchy.)

[jsfisher]

In other words, if we have a way to construct {}, {a,b,c,d,...} and anything between them, then we get P(A), and this is exactly what Cantor did by using: $$ D = \{ \, x \in X: x \not \in f (x) \, \} $$


I do note that your copy/splicing of the posts of others has lead you to notational inconsistencies. Can you find them?

ETA: By the way, are you ever going to define f(x) with other than just hand waving?

[doronshadmi]

It all makes sense now! Doron does his/her math without the use of thoughts!!

No at all.

It is both the simplest state of your Consciousness (which is not any thought) as the natural common ground of any thought or any relation among thoughts.


People like drkitten will continue to close your Third Eye, because they do not aware of the natural common ground of any thought (the simplest state of your Consciousness) but the sad joke is that they use this common ground as their hidden assumption, because no thought (definition) is the natural common ground of your Consciousness, but only some particular expression of it.


Drkitten's reasoning (which is currently and temporally the majority) is going to be obsolete soon, because it is an anti-evolutionist method that tries to address the naturally total by the naturally expressed, open, non-total and ever changing.

[jsfisher]

In other words, if we have a way to construct {}, {a,b,c,d,...} and anything between them, then we get P(A), and this is exactly what Cantor did by using: $$ D = \{ \, x \in X: x \not \in f (x) \, \} $$


(Emphasis added.) No, Cantor did no such thing. You said this before; you were corrected; repeating it is just a lie on your part. Stop lying.

Cantor needed exactly one set. Cantor defined exactly one set. One set (period).

[doronshadmi]

ETA: By the way, are you ever going to define f(x) with other than just hand waving?
Hand waver is a person that is not aware of how two thoughts of him are gathered into a one idea in his mind.

Can you tell us something about it jsfisher, or you still sell us your overused story about the definition that is the glue that gathers two other definitions into a one idea?

[doronshadmi]

(Emphasis added.) No, Cantor did no such thing. You said this before; you were corrected; repeating it is just a lie on your part. Stop lying.
In your dreams I was corrected. You are a liar of your own mind because you are not aware of the common ground of it and force completeness on the level of thoughts (the level of the expression of your mind, which is naturally non-total and ever changing).

Cantor needed exactly one set. Cantor defined exactly one set. One set (period).
Yes I know, the set of infinitely many distinct D versrions that exist between D={} and D={a, b, c, d, ...} (including D={} and D={a, b, c, d, ...}), that according to his construction method its cardinality cannot be but |Power(X)| > |X|.

No proof is nedeed here.

[doronshadmi]

"Simplicity itself!"

Ye, the glue between any two objects (abstract or not) which is something that cannot be known by your noisy mind.

For example http://forums.randi.org/showpost.php?p=3647417&postcount=70 is not in the image of your noisy mind, isn't it?

[Modified]

What makes this great is all the whining I've ever done about graders requiring me to be very exact in my proofs and what degree of exactness I need and when can I just say, "It's trivial."

I <3 the comedy gold going on here.

I once was talking to someone and it came up that he was in a class I was grading. When I told him he said "Oh, yeah... 'This is not a proof'". It was a standard comment of mine.

[jsfisher]

In your dreams I was corrected. You are a liar of your own mind because you are not aware of the common ground of it and force completeness on the level of thoughts (the level of the expression of your mind, which is naturally non-total and ever changing).

The easiest way to prove your are not a liar is to show exactly where in Cantor's proof he did what you say. Good luck with that, because he didn't.

Yes I know, the set of infinitely many distinct D versrions that exist between D={} and D={a, b, c, d, ...} (including D={} and D={a, b, c, d, ...}), that according to his construction method it cardinality cannot be but |Power(X)| > |X|.

No proof is nedeed here.

You cling to the "no proof is nedeed[sic]" only because you have no proof. Worse, you cannot even provide an explanation of how to construct P(X) from you collection of "D versrions[sic]". Why is that? Worse than worse, you cannot possibly show that your construction is identical to P(X) because you claim the definition of power set is unnecessary.

So, where does that leave you?

[Third Eye Open]

No at all.

It is both the simplest state of your Consciousness (which is not any thought) as the natural common ground of any thought or any relation among thoughts.

The simplest state of consciousness is having no thoughts...in other words being in a coma or dead. I guess I could agree with you there.


People like drkitten will continue to close your Third Eye, because they do not aware of the natural common ground of any thought (the simplest state of your Consciousness) but the sad joke is that they use this common ground as their hidden assumption, because no thought (definition) is the natural common ground of your Consciousness, but only some particular expression of it.

Now you lost me.
Sorry if I got your hopes up with my alias, but I'm actually just a big Tool fan. As far as I understand the word 'consciousness', having no thoughts would be the opposite of that, as in being unconscious.


Drkitten's reasoning (which is currently and temporally the majority) is going to be obsolete soon, because it is an anti-evolutionist method that tries to address the naturally total by the naturally expressed, open, non-total and ever changing.

I highly doubt that. If you make it a habit to force thoughts out of your head, I am sure it will be you who becomes obsolete. Probably because you didn't think the thought that you need to stop at a red light, or something.

[nathan]

You even do not aware of how two thoughts of you a gathered into a one idea, so go claim on another tree, where a definition (which is nothing but a thought) is the basis of your "closed under thoughts" circular reasoning.

I cannot understand this sequence of words. It starts with a capital letter, ends with a full stop and contains only words from the English language. Unfortunately it then diverges from usual sentence form.

Have you ever considered using simpler sentences? Then they might be grammatically correct at least, which might make it easier to understand what you're trying to say. (If it helps, consider writing as-if your target audience is not a native English speaker.)

[doronshadmi]

The simplest state of consciousness is having no thoughts...in other words being in a coma or dead. I guess I could agree with you there.
When you are aware of yourself without any thought, you are at the simplest state of your consciousness. A sea is still sea even if no wave (some thought) can be found, and this calm sea is the source of any wave (thought), and not vice versa.

You try to get this simplicity by making waves and from this noisy level you interpret this simplicity as no-consciousness or death.


Here is some analogy (and do not mix between the analogy in the real thing, because any definition, analogy, thought is nothing but some limited wave in the sea of consciousness):



The simplest state of consciousness is:
..._______________________________________...



Thoughts are:

/\ |
…____|_____/\____/ \____|_____________…


The waves (thoughts) are inseparable aspect of your consciousness exactly as a wave is inseparable of the sea, but since any wave is naturally limited, then any notion that is based on some wave (some thought) is naturally limited and cannot get the totality (completeness) of the sea as the natural source of any wave (thought, definition, etc. …).


If you try to get it only from the level of waves (thoughts) you will get a gap between any two thoughts that is experienced from the limited noisy state of waves (thoughts) as nothing, or no-consciousness or death, for example:

/\ nothing |
…____|_____/\____/ \________|_____________…



At the moment that the consciousness is aware of itself right from its simplest state (which is the sea without waves (thoughts)) and it is able to be aware of any thought as some creative expression of itself, then the consciousness is experienced as what it really is: the source of life, the source of action and re-action, the source of Logic, the source of abstract\non-abstract things and their infinitely many relations, that are always connected by the simplest state of the non-local atomic and naturally undefined (non-limited by any definition) unified sea of consciousness.


/\ toughts |
…____|_____/\____/ \________|_____________…

simplicity

[jsfisher]

So, where are we now? Let's recap:

Cantor's Theorem states the cardinality of the power set of a set A is strictly greater than the cardinality of the set A. Georg Cantor proved his theorem by an eloquent "diagonal method" argument.

The essence of the proof is to assume an onto mapping, f(x), exists between members of A and its power set, P(A). Then construct the set D of all the members, a, of A not included in f(a). Finally, show that D must be a member of P(A), but D is not covered by the mapping, f(x). Therefore, no onto mapping exists.


Doron has claimed


Cantor didn't use the meaning of "power set" in asserting the existence of the mapping, f(x).
It of course does. The mapping is from the members of set A and the power set of A. Clearly, the meaning of "power set" is incorporated into the definition of f(x).

The mapping f(x) was really just any mapping from members of the set A and a subset of A.
Cantor's proof uses words that say otherwise.

Cantor's proof relies on a "collection of D versions", a collection that presumably results from all the possible mappings, but doron isn't clear on this point..
It does not. One and only one set D is constructed by Cantor using one and only one f(x).

Cantor's proof then defines "power set" by constructing the power set for set A from the "collection of D versions".
It does not. The proof doesn't define the term, power set; that's a given at the start of the proof. It doesn't have a "collection of D versions" to work with, either.

The construction of the power set provided in the proof is such that cardinality of the power set of A simply must be strictly less than the cardinality of A. No proof is actually needed because it is just so obvious.
Even if the proof had defined power set by a construction it didn't provide from a "collection of D versions" that it didn't establish using all possible mappings from a set to any of its subsets that it didn't use, you'd still need to offer something more substantial than "no proof is necessary" before jumping to QED.


As has been her style, these are things doron simply asserts. No evidence is necessary. For that matter, no evidence is possible because doron is fabricating things in direct contradiction of any facts.

[nathan]

When you are aware of yourself without any thought, you are at the simplest state of your consciousness.
...
Oxymoronical bad analogy time.

Derail noted. How about actually proving your contention that Cantor's proof is overcomplicated?

ETA:jsfischer got there first :)

[Apathia]

When you are aware of yourself without any thought, you are at the simplest state of your consciousness.

Yes, No-Thought, Simple Awareness:
I've been there plenty of times.
But how is this your "Hidden Assumption?"
It makes no assumptions at all.
It seems to me you are trying to assign logic values to it, but you can't do that. It's prior to logic, period.
It's not "Non-Locality" as opposed to "Locality." Those are words denoting concepts. You cannot make Awareness an axiom of any system.
It doesn't excuse you making undocumented leaps in a mathematical demonstration.
It isn't any conceptual content or a free concept you can make anything you please.
Once the words start, it's gone. After a moment of pure concept free awareness, you may say, for example that there is no Completeness without the Incomplete, or Incompleteness without the Complete. All such concepts interpenetrate each other.
And you can make a fuzzy, non-ecluded Logic to express the Coincidence of Opposites.
But that doesn't excuse sloppy mathematics,
And it doesn't grant you the criticism of Cantor you think it does.

Is the "Coincidence (or interpenetration) (or mutual interdependence of Opposites)" your "Hidden Assumption?"
What you are trying to do with this concept does not follow.
Ironically Transfinite Numbers are possible the Infinite is not an Absolute content.

[Apathia]

Ironically Transfinite Numbers are possible because the Infinite is not an Absolute content.

Too late to edit this out. I'd assert it philosophically, but I don't have the mathematically tools needed to demonstrate it. So nevermind I said it.

My main point is that playing the mystic card isn't giving you the winning hand you think it is. (Especially to an audience that is decidedly non-mystic.)

[doronshadmi]

jsfisher,

No proof is needed here!

Your unnecessary complicated maneuvers do not hold water, exactly as the claim that is the expression 0 < x < 1 there is no grantee that x is between 0 and 1.

[doronshadmi]

You cannot make Awareness an axiom of any system.

You have missed it.

The right one is this: You cannot get any axiom without Awareness.

[jsfisher]

jsfisher,

No proof is needed here!

Yes, I understand. It is unfair of me to ask you to do something you cannot possibly do.

That aside, though, I wasn't really asking you to prove anything, at least not yet. You have made claims with regard to the proof of Cantor's Theorem. All I am asking for is where in the proof are these things you claim are there?

Is that really too much to expect?

Your unnecessary complicated maneuvers do not hold water

I did nothing more than recite your claims. I agree your claims seem to involve unnecessarily complicated maneuvers, but that's not my doing.

...exactly as the claim that is the expression 0 < x < 1 there is no grantee that x is between 0 and 1.

I didn't make such a claim. Did you? Be that as it may, what does that have to do with Cantor's Theorem?

[Apathia]

You have missed it.

The right one is this: You cannot get any axiom without Awareness.

Couldn't agree more.
My point is that Awreness is not objective information you can plug into an axiomatic system.
It's not intellectual content.
You are confusing your mathematical intuitions with Awareness.
This is a big no-no.

You are certainly free to state your definitions and axioms and construct your own formal system. It may be different from standard mathematics.
It's not like there is a "true" Mathematics. anymore than there is a "true" language. You say your system has certain advantages. Just what are those?

[doronshadmi]

Oxymoronical bad analogy time.
No, without the simplest state of your consciousness no two thoughts can be connected with each other.

The same holds for Logic where True\False cannot be connected with each other, and this is also the case of any collection of abstract\non-abstract objects.

The relater can be used as a hidden assumption of the related, but it cannot be avoided.

It is fundamental an inevitable, you are using it as a hidden-assumption and I explicitly use it.

When explicitly used, no collection or related things is total when compared to the relater.

It is simple beautiful and logically addressed in http://www.geocities.com/complementarytheory/NXOR-XOR.pdf .

I did my best in order to explain it, but there is a limit to any explanation because the explained is never the thing itself. Since you have never directly experienced awareness without thoughts, there is nothing to talk about.

If you wish to get such an experience, then, for example, go and learn Transcendental Meditation or any other similar meditation technique.

[jsfisher]

Can we please keep the derails and spamming to a minimum?

The subject of this thread is Cantor's Theorem, as interpreted by Doron Shadmi.

Please save the "hidden assumption", metaphysics, and other buzz-killing discussions for other threads.

[Apathia]

When explicitly used, no collection or related things is total when compared to the relater.

This is precisely where you lose it. You make an objective concept out of that which isn't conceptual and isn't objective.
That confusing concept then becomes an obstruction for you.

[doronshadmi]

Couldn't agree more.
My point is that Awreness is not objective information you can plug into an axiomatic system
Apathia,

Mathematics is non-researchable\researchable complementation, where the non-researchable is the natural glue of any two researchable things.

When you get that, then and only then there are no hidden-assumptions at the basis of your reasoning.

You say your system has certain advantages. Just what are those?
It can be used as a common framework for both Ethics and Logic.

The dichotomy of the current paradigm easily enables technical developments that are not based on the simplest state of consciousness, which is the non-personal aspect of your and my personal states of mind.

This non-personal aspect is a universal constant of any thought, exactly like the speed of light in Einstein's SRT.

If any personal awareness is tuned to act according to the universal constant of the non-personal simplest state of awareness, then the interactions between persons and between persons and their near and far environment, will not contradict each other because the technology of the consciousness will be the natural basis of any other technology.

Let us say that if the technology of the consciousness will not be in the very near future the fundamental technology of our species, we shell not survive the increased powers of nature that are discovered and manipulated by our personal-only awareness, which its limitation is the cause of the current dichotomy between Ethics and Logic, end this dichotomy will lead us to some Evolution's dead end street.

[doronshadmi]

This is precisely where you lose it. You make an objective concept out of that which isn't conceptual and isn't objective.
That confusing concept then becomes an obstruction for you.
No, the non-personal aspect of your consciousness is actually the most objective thing. Any thought (definition) is forced to be "objective" by some agreement between community of persons, but it is just an illusion of objectivity.

[Apathia]

Apathia,

Mathematics is non-researchable\researchable complementation, where the non-researchable is the natural glue of any two researchable things.

When you get that, then and only then there are no hidden-assumptions at the basis of your reasoning.

It can be used as a common framework for both Ethics and Logic.

The dichotomy of the current paradigm easily enables technical developments that are not based on the simplest state of consciousness, which is the non-personal aspect of your and my personal states of mind.

This non-personal aspect is a universal constant of any thought, exactly like the speed of light in Einstein's SRT.

If any personal awareness is tuned to act according to the universal constant of the non-personal simplest state of awareness, then the interactions between persons and between persons and their near and far environment, will not contradict each other because the technology of the consciousness will be the natural basis of any other technology.

Let us say that if the technology of the consciousness will not be in the very near future the fundamental technology of our species, we shell not survive the increased powers of nature that are discovered and manipulated by our personal-only awareness, which its limitation is the cause of the current dichotomy between Ethics and Logic.

Doron,

I'm all for a transpersonal perpective, but your Logic isn't a "Technology of Consciouness" and neither serves compassionate awareness or does diservice to it.
I see your issue with Cantor isn't really a mathematical but a religious one.

But I know from hard experience not to bother to place myself between someone and their religious assertions.
And I'm off topic anyway.

Sorry, jsfisher.
Since I don't have a mathematical contributionj to make here, I'll bow out.

Doron, Spirituality isn't a Logic.
Awareness isn't an assumption.
Transcend your Logic into Compassion.

[jsfisher]

Sorry, jsfisher.
Since I don't have a mathematical contributionj to make here, I'll bow out.

Your perspective is most interesting, Apathia. You are much better versed in the philosophical arts than I, but I can still appreciate the elegance of what you are saying.

Still, my concern, here, and thus my plea for fewer derails, is that doron has a strong tendency to obfuscate things when anyone tries to pin down something she's said. She's used secret word definitions and out-right gibberish in explanations, bald insults towards anyone that isn't fluent in gibberish, and then excursions into metaphysical debates to blur the original points.



So, doron, please point out in the proof of Cantor's Theorem all these things you claim are there.

[Apathia]

Still, my concern, here, and thus my plea for fewer derails, is that doron has a strong tendency to obfuscate things when anyone tries to pin down something she's said. She's used secret word definitions and out-right gibberish in explanations, bald insults towards anyone that isn't fluent in gibberish, and then excursions into metaphysical debates to blur the original points.

Yes, keep pushing Doron to provide mathematical clarity and precision.
It appears to me that hir misunderstandings arise more from how s/he understands sets and number than the mystical stuff. For example s/he's very strong on the notion of set as a container even though s/he gives a superficial definition as a collection.

I'll still be lurking. It's been a good learning experience for me.

[Reality Check]

doronshadmi:
Since you are obviously an expert on Cantor's Theorem (http://en.wikipedia.org/wiki/Cantor%27s_theorem), you will be able to easily give an answer to a question that is puzzling me:

What book, article or other publication gives the proof of Cantor's Theorem using a collection of D versions where D is you have given it
This is the subset B as given in the Wikipedia article

Let f be any function from A into the power set of A. To establish Cantor's theorem it must be shown that f is necessarily not surjective. To do that, it is enough to exhibit an element of the power set of A, that is, a subset of A, that is not in the image of f. Such a subset can be constructed as follows. Take subset B of A defined by:
http://upload.wikimedia.org/math/f/d/c/fdc55bfbf098055688fc13e561e572cf.png
This means, by definition, that for all x in A, x ∈ B if and only if x ∉ f(x), so for all x the sets B and f(x) are different. There is no x such that f(x) = B; in other words, B is not in the image of f.

[doronshadmi]

I see your issue with Cantor isn't really a mathematical but a religious one.
No, it is a matter of consciousness, the exact science of consciousness.

No woo woo or any other mystical approach is involved here.

It is about time that consciousness will be directly understood in terms of a rigorous framework based on scientific methods and my suggested framework is exactly the first step to this direction.

The very first step is to directly and unconditionally understand how two thoughts are gathered into a one idea. If you rigorously get that you immediately understand how D versions are exactly Power(X).

[doronshadmi]

superficial definition as a collection.
Superficial understanding of a collection is any non-ability two rigorously show how two thoughts are gathered into a one idea. In other words, this is exactly the situation of the current paradigm of the mathematical science.

[doronshadmi]

She's used secret word definitions
On the contrary.

I use the simplest definitions that are based on the simplest reasoning.

Your mystical point of view about anything that is related to the concept of consciousness is your problem, not mine.

Furthermore the religious\mystical\woo woo point of view of consciousness is the most irrational and dangerous point of view of our species of this most fundamental thing, and we immediately have to do our best in order to scientifically use it in our daily life, before it is too late and our current religious\mystical\woo woo point of view of consciousness will surly lead us into an Evolution's dead end street.

[Reality Check]

doronshadmi, citations?

[doronshadmi]

doronshadmi, citations?

You will not find it (yet) on any Book or article, because I am the first person that shows it.

No book is needed here if you understand How Ds (or B of the Wiki version) distinct versions are exactly the infinitely many Power(X) members, where each one of them is constructed in such a way that is always beyond the image of any given X member (and the result cannot be but |Power(X)| > |X|).

Persons that belong to the current community of mathematicians will try to do their best in order to save the complicated and un-necessary "proof" by contradiction of Cantor, because it is one of their fundamental agreed terms that save the stability of this community. For example see jsfisher and drkitten that when asked how two thoughts are gathered into a one idea, will do any possible maneuver in order to avoid the answer, because if the answer is given, the result of Ds as Power(X) members that are also beyond the image of and X member, is simply inevitable.

It is simple, beautiful and no proof is nedeed here, because Cantor directly constructs it.

[doronshadmi]

So, doron, please point out in the proof of Cantor's Theorem all these things you claim are there.

It cannot be seen by persons that cannot answer to this question:

How two thoughts are gathered into a one idea?


( http://forums.randi.org/showpost.php?p=3652719&postcount=137 , http://forums.randi.org/showpost.php?p=3647417&postcount=70 )

[doronshadmi]

Ok, so what? 2, 3 and 4 are each members of N, but that doesn't define the set N. Your D sets do not define P(X) either.

This is not the Ds case, because we have D={}, D={a,b,c,d,...} and any D version between them, where each one of them is a Power(X) member that is out of the image of (it is not mapped with) any X member.

This is simply and directly defined by

$$ D = \{ \, x \in X: x \not \in f(x) \, \} $$

and no proof by contradiction is needed here.

Also it is "nice" that you ignored http://forums.randi.org/showpost.php?p=3650854&postcount=117 .

[nathan]

If you wish to get such an experience, then, for example, go and learn Transcendental Meditation or any other similar meditation technique.

So this has nothing to do with your argument about Cantor's proof. Could you perhaps stick to the point you originally made. If you want to discuss metaphysics, I believe you know where to go.

[jsfisher]

You will not find it (yet) on any Book or article, because I am the first person that shows it....It is simple, beautiful and no proof is nedeed here, because Cantor directly constructs it.


In other words, you cannot show where Cantor constructs the power set because he doesn't. You made the whole think up. You cannot show us how Cantor constructs the power set because it is all a lie.

[jsfisher]

This is not the Ds case, because we have D={}, D={a,b,c,d,...} and any D version between them, where each one of them is a Power(X) member that is out of the image of (it is not mapped with) any X member.


Only in your misguided imagination. The proof of Cantor's Theorem constructs no such sets. You are a liar to state otherwise. The actual proof constructs one and only one set based on one and only one mapper between X and P(X).

This is simply and directly defined by

$$ D = \{ \, x \in X: x \not \in f(x) \, \} $$

Yes, that is correct. That defines one set D based on that singular f(x).

and no proof by contradiction is needed here.

If that is your belief, then you clearly don't understand the concept of Mathematical proof.

Also it is "nice" that you ignored http://forums.randi.org/showpost.php?p=3650854&postcount=117 .

I didn't ignore it; it deserved no response. It was just more of your repetitive nonsense that shows a basic misunderstanding for obvious.

[Apathia]

Doron has made his conceptions regarding Consciousness critical to his "proof" in this thread. So I'm going to point out where the disconnect begins.

Doron says:
In this case we have no choice but to use an analogy, but we must not mix between simplicity itself (which is naturally undefined) and some analogy of it (which is limited like any other definition).

Precisely. Of the inneffable, we can only make analogies. The analogies can serve as expression for communication, but they always remain merely the map, not the territory.
Doron attempts to express the relationship between Thought and No-Thought (No-Thought being the state of consciouness in which there is just awareness without thoughts and boundaries) in a mathematical notation where the ineffable ground is given logic values and treated as a predicate.
The map gets quickly confused for the territory. The concrete experience of consciousness is mistaken for an intellectual concept and mathematical abstraction. It becomes a single tier of predicated values in hir logic.

At just the point Awareness is made a "new magnitude," the fallacy begins.

Then s/he runs with it and attaches other unstated concepts in undocumented leaps to dubious conclusions.
When those assuptions are questiond, she attraches them to "consciousness," and claims they would be apriori if we were self-aware.

If we were to take Doron's logic to computer science, programming that tier of logic value s/he uses to represent consciouness, we'd not get a self-conscious AI entity. At best we'd have some fuzzy systems. But we already have mathematics of a non-excluded middle. Obviously consciousness is something other than this mere stick drawing.

By analogy you can call Consciouness a "common ground" to human mental activity. But I'd like to submit that "groundlessness" is probably a more apt analogy.

In catagorizing consciousness as an intellectual concept, something gets lost in the translation. Just like the map can't fully express the territory.
In the Doron's system this shows up as a denial of a robust mathematical infinity.
Of course, thoughts can't be complete, but thoughts don't have the last word, unless you are confusing concepts for reality.

I apologise for Philosophy in a Mathematics thread. But this is where Doron keeps running and running aground.
Doron will continue to say we don't get it because we haven't experierenced Consciousness.
And as someone who has experienced "No-Thought" states of consciousness, I protest hir confusing a conceptual wire rack for the inconceptual and trying to fill that which is empty of concept with intelectual analogies.

[Reality Check]

You will not find it (yet) on any Book or article, because I am the first person that shows it.

No book is needed here if you understand How Ds (or B of the Wiki version) distinct versions are exactly the infinitely many Power(X) members, where each one of them is constructed in such a way that is always beyond the image of any given X member (and the result cannot be but |Power(X)| > |X|).

Persons that belong to the current community of mathematicians will try to do their best in order to save the complicated and un-necessary "proof" by contradiction of Cantor, because it is one of their fundamental agreed terms that save the stability of this community. For example see jsfisher and drkitten that when asked how two thoughts are gathered into a one idea, will do any possible maneuver in order to avoid the answer, because if the answer is given, the result of Ds as Power(X) members that are also beyond the image of and X member, is simply inevitable.

It is simple, beautiful and no proof is nedeed here, because Cantor directly constructs it.

You shows what?

Please show us where in Cantor's proof there is a "collection of D versions" or many "D distinct versions". Then prove that these versions form a power set.

PS. Cantor's proof is not complex. It is simple, concise and necessary.

[doronshadmi]

At just the point Awareness is made a "new magnitude," the fallacy begins.
1. It is not a new magnitude, but it is Totality.

2. At the moment that it is understood, then anything that is not-total is naturally incomplete, and a collection is a non-total (and incomplete) thing, no matter what magnitude it has, this magnitude is incomplete.

3. When the difference between the Total (and non-researchable) and the non-total (and the researchable) is understood, then the "closed under thoughts" fallacy gets off stage, and the organic paradigm of Total\Non-total complementation, which is a naturally not-circular framework, gets on stage.

4. I clearly and simply provided the logical aspect of the organic paradigm of the mathematical science ( http://www.geocities.com/complementarytheory/NXOR-XOR.pdf )
and the "closed under thoughts" circular fallacy is not relevant anymore.

[doronshadmi]

Please show us where in Cantor's proof there is a "collection of D versions" or many "D distinct versions". Then prove that these versions form a power set.

PS. Cantor's proof is not complex. It is simple, concise and necessary.

The result of |Power(X)| > |X| is directly given by D construction, or in other words, any "proof" here is more complicated than Ds direct construction.

[Reality Check]

The result of |Power(X)| > |X| is directly given by D construction, or in other words, any "proof" here is more complicated than Ds direct construction.
Pleae give us the steps in your "D construction" and map them to Canot's proof.

[doronshadmi]

Pleae give us the steps in your "D construction" and map them to Canot's proof.
http://forums.randi.org/showpost.php?p=3644296&postcount=39
http://forums.randi.org/showpost.php?p=3654898&postcount=161

[Reality Check]

http://forums.randi.org/showpost.php?p=3644296&postcount=39
http://forums.randi.org/showpost.php?p=3654898&postcount=161

I read those and they have nothing to do with Cantor's Theorem. D in the proof is a single subset and not a collection of subsets.

Where in the proof of Cantor's Theorem is this collection of D variants?

If it is your own proof then why mention Cantor at all. Just publish your proof.

[doronshadmi]

I read those and they have nothing to do with Cantor's Theorem. D in the proof is a single subset and not a collection of subsets.
1. It is not Cantor's theorem.

2. D is not a single subset, but it is a direct construction of each of Power(X) members in such a way that any D version is out of the range (it cannot mapped with) of any X member.

3. It is not a proof, but it is a direct construction that its result cannot be but |Power(X)| > |X|.

4. Any proof of this case is nothing but an unnecessary twisted maneuver.

Please look at this: http://www.tcs.tifr.res.in/~raja/publications/online/yapct08.pdf .

The mistake of this proof without negation is based on the term "all" (completeness) of a collection, but since a collection is an incomplete object, also this proof does not hold.

[jsfisher]

1. It is not Cantor's theorem.

You started off this thread claiming it to be about Cantor's Theorem and its proof. But, if that's not what it is about, please tell us what it really is.

2. D is not a single subset, but it is a direct construction of each of Power(X) members in such a way that any D version is out of the range (it cannot mapped with) of any X member.

This is certainly not the case in the proof of Cantor's Theorem; however, since you seem to have jumped topics, where does this D set of which you speak arise?

3. It is not a proof, but it is a direct construction that its result cannot be but |Power(X)| > |X|.

It is definitely not a proof of anything. Moreover since you have been very frugal with details, there is nothing even suggesting a conclusion about cardinality in anything you have provided.

4. Any proof of this case is nothing but an unnecessary twisted maneuver.

Your statement speaks more of its author's ability to understand the proof than the proof itself.


So, the bottom line in all this is you malign Cantor, then refuse to support any of your claims.

[Apathia]

1. It is not a new magnitude, but it is Totality.

2. At the moment that it is understood, then anything that is not-total is naturally incomplete, and a collection is a non-total (and incomplete) thing, no matter what magnitude it has, this magnitude is incomplete.

3. When the difference between the Total (and non-researchable) and the non-total (and the researchable) is understood, then the "closed under thoughts" fallacy gets off stage, and the organic paradigm of Total\Non-total complementation, which is a naturally not-circular framework, gets on stage.

4. I clearly and simply provided the logical aspect of the organic paradigm of the mathematical science ( http://www.geocities.com/complementarytheory/NXOR-XOR.pdf )
and the "closed under thoughts" circular fallacy is not relevant anymore.

I suppose it's possible that you have something to say, and I'm just misunderstanding you at every turn. But every opportunity you have to steer me on your path, you throw in a new, unexplained turn of phrase.
This time it's "closed under thoughts circular fallacy."
All I know of this is that it must be something I have.

But fairly speaking, I'm replying to you with terms other than ones you ordinarily use. So we aren't getting anywhere.

However your #2 point above is pretty clear for the layman and novice in Mathematics. And heck, you can do your own system where your intuitions involved in this fully hold. You are right to call it a different paradigm, because makes a really substantial departure from Mathematics as we know it.
All of the mathematics contemporary Physics is based upon from Newton to Hawking has to be scrapped, and you need to start over with something else.
Unfortunately it goes back further than that. We are back with Zeno of Elea still confused over the relationship between the continuous and discrete and number and line. Oh yes, you do have a new approach, a new path. Good luck. But realize you are setting aside and ripping through more in modern Mathematics and its scientific applications than you realize.
There's much more at issue than just the work of Cantor.

Which is why professional Mathmaticians hardly even bother to discuss these things with you and you:
At this stage I prefer to share my notions with non-professional mathematicians, because they are immediately rejected by professional mathematicians for a good reason (from their point of view) which is: I add a new magnitude that if it is logically understood, then any given non-finite collection is incomplete when compared to the new magnitude.

It seems to me that there is something positive in your attitude and the direction you want to go. I could be wrong about this, but I see you trying to make a philosophical shift away from absolutes to dynamic relation. It shows in your preferance for a multi-valued logic and your attempts at Complementarity. I keep reading your stuff just on the possibility that there is something in your approach that substantally serves this. I'm not finding it yet, but I can see how its a valuable stage in your quest.

Just beware: The Tao that can be expressed as a tier in a truth table, is not the Tao.
Oh wait! That must be my "closed under thought fallacy!"
Of course thought can have symbols for Totality, and even for the tottaly complete.

I need leave this thread to the mathematicains. I've participated in too much off-topic, though not as much as Doron hirself.

[doronshadmi]

However your #2 point above is pretty clear for the layman and novice in Mathematics.


No, my #2 point is pretty clear that no current mathematician knows how two thoughts are gathered into a one idea, or in other words, the current mathematical science is a circular "closed under thoughts" framework.

Try to understand it by thought and at this very moment you do what is needed in order to not understand the Total\Non-total organic framework.

This mistake is clearly shown at the basis of Geometry (See Hilbert's axioms and his fallacy that is based on defining a non-local atom, called a line, by local atoms, called points), Set theory (a collection of distinct objects is complete), Number theory (there is no general understanding of this concept), Logic (There is no answer of how at least two opposites are simultaneously connected without immediately contradicting each other).

As a current mathematician you do not have any basis in order to get the organic paradigm, unless you are not "closed under thoughts".

Of course thought can have symbols for Totality, and even for the tottaly complete.
No thought about Totality or some symbol of it, is Totality at its self state.

[nathan]

this thread needs moving to religion and philosophy. Doron is showing no signs of doing any maths.

[drkitten]

this thread needs moving to religion and philosophy. Doron is showing no signs of doing any maths.

She's showing no signs of doing any religion or philosophy, either.

[Don At Work]

Yes I was refuted by people like you that are unaware of the common basis of their thoughts, which is itself not a thought, but the simplest state of consciousness, which is the natural basis of any thought (and since a definition is some thought, it is also the natural basis of any definition).

Also a Computer stuff (or any other abstract or non-abstract mechanical method) is nothing but some agent of your consciousness.

In other words, real mathematician is first of all a person that is aware of the simplest state of consciousness as an inseparable part of his mathematical work.


Ok. You win. This is not even wrong. In the same light, this is not even Woo.
Since I have NO IDEA what you are on about, here is a bunny with a pancake on it's head. :bunpan

[Apathia]

No, my #2 point is pretty clear that no current mathematician knows how two thoughts are gathered into a one idea, or in other words, the current mathematical science is a circular "closed under thoughts" framework.

Try to understand it by thought and at this very moment you do what is needed in order to not understand the Total\Non-total organic framework.

This mistake is clearly shown at the basis of Geometry (See Hilbert's axioms and his fallacy that is based on defining a non-local atom, called a line, by local atoms, called points), Set theory (a collection of distinct objects is complete), Number theory (there is no general understanding of this concept), Logic (There is no answer of how at least two opposites are simultaneously connected without immediately contradicting each other).

As a current mathematician you do not have any basis in order to get the organic paradigm, unless you are not "closed under thoughts".

No thought about Totality or some symbol of it, is Totality at its self state.

Sorry Doron, I'm not able to give myself a mathematical developmental disorder at this time.

[doronshadmi]

Sorry Doron, I'm not able to give myself a mathematical developmental disorder at this time.
Right now you are in the middle of a mathematical developmental disorder because you a using a "closed under thouhgts" circular reasoning.

[doronshadmi]

this thread needs moving to religion and philosophy. Doron is showing no signs of doing any maths.
On the contrary.

You have a religious approach of the mathematical science, because you unable to get the simple notion that ideas can be changed by a paradigm-shift (exactly as some mutation changes some biological system from within).

You do not understand that the organic framework of the mathematical science is the real rigorous framework because it is not a circular "closed under frozen thoughts" framework.

[doronshadmi]

She's showing no signs of doing any religion or philosophy, either.
Only the most exact science, The rigorous science of the consciousness, which is the anti-thesis of the woo woo mystical\religious approach of this most important concept.

[doronshadmi]

Ok. You win. This is not even wrong. In the same light, this is not even Woo.
Since I have NO IDEA what you are on about, here is a bunny with a pancake on it's head. :bunpan

Don At Work they are beautiful
:bunpan :bunpan :bunpan :bunpan :bunpan :bunpan :bunpan :bunpan :bunpan
:bunpan :bunpan :bunpan :bunpan :bunpan :bunpan :bunpan :bunpan :bunpan

[drkitten]

Only the most exact science, The rigorous science of the consciousness,

The "rigorous science of the consciousness."

The cool of the summer sun. The calm of a category 5 hurricane. The stealthy approach of a stampeding herd of rhinos. The fiscal responsibility of a drunken sailor on leave.

And the writing precision of doronshadmi.

[doronshadmi]

You started off this thread claiming it to be about Cantor's Theorem and its proof.
Yes.

No proof is needed here, because we have a direct construction of the result, in this case.

Let alone show your comparison between N (which is not closed from above, for example 1,2,3,4,…) http://forums.randi.org/showpost.php?p=3654898&postcount=161 and R (which is both can be closed from below and above, for example [0,1]) in order to conclude that you do not understand this subject.

If you did not get that yet then x of 0 =< x =< 1 is equivalent to D of {{}, … , {a, b, c, d, …}}.

[nathan]

You have a religious approach of the mathematical science, because you unable to get the simple notions that ideas can be changed by a paradigm-shift (exactly as some mutation changes some biological system from within).

A major shift in ideas requires *evidence*, something you have failed to provide.

You do not understand that the organic framework of the mathematical science is the real rigorous framework because it is not a circular "closed under frozen thoughts" framework.

Mathematics is not an organic science. Organic science involves itself with the chemistry of carbon atoms at the lowest level, and moves upwards to encompass biology. You appear to be making a classification error here.

[doronshadmi]

The cool of the summer sun.
You are playing with words, but you are unable to answer to this question:

How two opposite terms are gathered by you in the first place?

As long as the answer is not given, you have exactly nothing to say about the organic framework.

[doronshadmi]

A major shift in ideas requires *evidence*, something you have failed to provide..
The *evidence* is exactly the simplest state of your consciousness when it is aware of itself without thoughts (it is not a thought about the simplest state, but it is the simplest state, which is naturally not a thought, but it is the source of any thought).

This is indeed a major shift for anyone how get things only from the level of thoughts.



Mathematics is not an organic science.
Think general. Organic means the result of the complementation between the non-local and the local.

[drkitten]

You are playing with words, but you are unable to answer to this question:

How two opposite terms are gathered by you in the first place?

Actually, I can answer that quite easily. Human memory, including vocabulary, is stored as a distributed pattern of neural activation, in such fashion that "thinking" of one concept, including reading it or looking at examples, will automatically "activate" other, related concepts. This is well-known and understood in the psych literature under the name "priming."

One easy and close relationship is contradiction; thinking of one concept will raise the activation level of the related-but-opposite concept (i.e. thinking of heat or of hot things will raise the mental salience of "cold" and related words.) So opposites are trivial. If you want the math, Hertz, et al. have a very good description of the mathematics of various models of associative memory.

This is also supported by the studies of linguistic cooccurance statistics (without reference to specific neural models), such as the LSA model developed at the University of Colorado.

Now that I've answered your question and supported it with references to the literature, let me ask you one.

How are totally unrelated gibberish terms gathered by you?

[nathan]

The *evidence* is exactly the simplest state of your consciousness when it is aware of itself without thoughts (it is not a thought about the simplest state, but it is the simplest state, which is naturally not a thought, but it is the source of any thought).

I see you're still not attempting to write simpler sentences. No matter. As has been pointed out having no thoughts and being self aware conflict. If one is self aware, one has a thought. If one has no thoughts, one is unconscious (and so not self aware).

This is indeed a major shift for anyone how get things only from the level of thoughts.
This is not a sentence. Consequently I don't know what you mean.

Think general. Organic means the result of the complementation between the non-local and the local.

Not in my dictionary it doesn't. My Concise OED gives:
1) of the bodily organs
2) having organs or organized physical structure
3) ... containing carbon in its molecule(s)
4) constitutional, inherent, fundamental structural
5) organized or systematic.

of those, I think 3, 2 and 1 (in that order) are what 'organic science' is understood to mean.

Where are you finding your definition of organic? Or did you just make it up?

Your previous post used the compound noun 'organic science', now you seem to have dropped the science part -- is that significant?

[doronshadmi]

Actually, I can answer that quite easily. Human memory, including vocabulary, is stored as a distributed pattern of neural activation, in such fashion that "thinking" of one concept, including reading it or looking at examples, will automatically "activate" other, related concepts. This is well-known and understood in the psych literature under the name "priming."

One easy and close relationship is contradiction; thinking of one concept will raise the activation level of the related-but-opposite concept (i.e. thinking of heat or of hot things will raise the mental salience of "cold" and related words.) So opposites are trivial. If you want the math, Hertz, et al. have a very good description of the mathematics of various models of associative memory.

This is also supported by the studies of linguistic cooccurance statistics (without reference to specific neural models), such as the LSA model developed at the University of Colorado.

Now that I've answered your question and supported it with references to the literature, let me ask you one.

How are totally unrelated gibberish terms gathered by you?


You describe results without show their basis.

You did not answer to the question, which is:

How two thoughts are connected with each other?

To say that x leads to y does not answer to this question, because x is a one thing and y is another thing, and no one of them is the connector and each one of them is a connected thing.

So what is the connector that connects x to y or y to x without eliminating their self identities during the connection?

Hilbert's Geometric axiomatic system is a good example of this misunderstanding, because he tries to define the connector (represented as Line) in terms of the connected (represented as points).

This fallacy is fundamental to the current mathematical science and appears in any given branch like Set theory, Logic, Number theory etc. …

[doronshadmi]

As has been pointed out having no thoughts and being self aware conflict.
No, thoughts are nothing but the expressed aspect of the awareness, where the awareness at its simplest state is directly aware of itself without any thought.

The simplest state is the non-personal state of any personal expression (including thoughts) exactly as the sea is the non-local state of every wave (which is a limited aspect of the unbounded calm sea).

If you do not understand that the researchable as the result of the synthesis between the total connector (Unity) and the totally disconnected (Isolation) you do not understand the essence of the Organic paradigm.

[Ateius]

Consider the following:

All men are mortal.
Socrates is a man.
The best flavor of pie is cherry.
Therefore, Socrates is mortal.

With all due respect, drkitten - and please know I think the utmost of your contributions to this forum, and the expansion in my own learning that they provide - I must disagree with you here.

The best flavour of pie is apple.

[jsfisher]

this thread needs moving to religion and philosophy. Doron is showing no signs of doing any maths.


Please note that it did get moved. Thank you, moderators.

Doronshadmi, your thread is no longer in the math and science forum mostly because you were/are unwilling to discuss any Mathematics. This forum is better suited to your ramblings, so enjoy yourself. As for me, though, this section of JREF is not my cup of tea, so so long and thanks for all the fish.

[nathan]

No, thoughts are nothing but the expressed aspect of the awareness, where the awareness at its simplest state is directly aware of itself without any thought.

The simplest state is the non-personal state of any personal expression (including thoughts) exactly as the sea is the non-local state of every wave (which is a limited aspect of the unbounded calm sea).

If you do not understand that the researchable as the result of the synthesis between the total connector (Unity) and the totally disconnected (Isolation) you do not understand the essence of the Organic paradigm.

More gibberish. anyway, I find most of the philosophy here akin to mental masturbation, so I'll leave you to your badly thought out ramblings.

[doronshadmi]

Please note that it did get moved. Thank you, moderators.

Doronshadmi, your thread is no longer in the math and science forum mostly because you were/are unwilling to discuss any Mathematics. This forum is better suited to your ramblings, so enjoy yourself. As for me, though, this section of JREF is not my cup of tea, so so long and thanks for all the fish.
Yes.

You are unable to answer to a fundamental question, which is:

How two thoughts (definitions) are connected with each other into a one idea?

The current mathematical science cannot answer to this question because it is a circular "closed under thoughts (definitions)" framework.

Some moderator gave a way to continue your dichotomic (it is not in the Mathematics forum, la la la ...) ignorance, isn't it?

[doronshadmi]

More gibberish. anyway, I find most of the philosophy here akin to mental masturbation, so I'll leave you to your badly thought out ramblings.
Go Nathan go and continue your circular "closed under definitions" masturbation. Any circular method is first of all a "full gas in neutral" masturbation, and you are a good example of this method, first of all because you are blind to your own consciousness.

[doronshadmi]

Please note that it did get moved. Thank you, moderators.

Doronshadmi, your thread is no longer in the math and science forum mostly because you were/are unwilling to discuss any Mathematics. This forum is better suited to your ramblings, so enjoy yourself. As for me, though, this section of JREF is not my cup of tea, so so long and thanks for all the fish.
So you avoid http://forums.randi.org/showpost.php?p=3660363&postcount=185 .

Please explain us how x of the expression 0 ≤ x ≤ 1 is one and only one case that is 0 ≤ and\or ≤ 1?

[doronshadmi]

cooccurance statistics (without reference to specific neural models)

Co-occurrence ( http://lsa.colorado.edu/papers/plato/plato.annote.html ) is exactly x-occurrence, y-occurrence and something that is called "Co-" that is not another occurrence, but it is the Connecter of x,y occurrences.

You cannot avoid this fundamental Organic building-block as the most primitive researchable thing, so LSA is another "closed under thoughts" circular reasoning that uses the "Co-" as its hidden assumption.

Each time that anyone uses lines and dots in order to represent his results by some diagram, the "Co-" is most of the time represented as a line-like element (a non-local element), and the occurrence is represented by a point-like element (a local element).

In other words, you did not provide the right answer. All you did is to continue to use the "Co-" as your hidden assumption.

[drkitten]

Co-occurrence ( http://lsa.colorado.edu/papers/plato/plato.annote.html ) is exactly x-occurrence, y-occurrence and something that is called "Co-" that is not another occurrence, but it is the Connecter of x,y occurrences.

More words you don't understand.


In other words, you did not provide the right answer.

You do not have sufficient understanding to make that pronouncement.

[doronshadmi]

More words you don't understand.

Said the man that does not able to know the basis of his own consciousness.